Hi there Simon, and Welcome to the forum!
I had originally intended to buy a large wooden shed, line the walls with some 'magic soundproofing materials' and have the job done over a weekend for around £2000. I then discovered this forum and realised I had a lot to learn!

Thankfully, you found us in time, and didn't make that huge mistake!
6 months later I'm now going with concrete walls and roof, and had to increase my budget ten fold. Thanks a lot!
Only ten-fold???

We must be doing something wrong, then...
The primary purpose is as a pleasant environment for composing as well as playing and recording piano and violin. The secondary purpose is as a control room for mixing. I appreciate that I will need to make some compromises as it can't really be a great live room and control room, but where those need to be made I would favour the live side of things. (I need enough floor space to be able to squeeze in a small drum kit and double bass now and then for a rehearsal so I don't want to split the space in to more than one room).
For such a broad and highly varied range of uses, I would strongly suggest that you consider doing variable acoustic treatment, so you can change the acoustic response of the room across a wide range, from "bright" to "dull", and "specular" to "diffusive". There's no way you are going to get one single set of treatment that works well for all of those. Rather, you'd get one single set of treatment that sounds lousy for all of them!
Around £20,000 (25,000 USD)
You have about 28m2 footprint, as near as I can tell, so you are planning on spending less than £ 1k per square meter. That's a little on the low side.... I have designed a few studios for the UK, some from the ground up (like yours), and the clients who have built their places have told me that the cost is generally around £ 1,000 to £ 1,500 /m2. You might want to increase your budget.
I'm in the UK, the studio is at the bottom of my garden, around 25m from the nearest house.
Given the dimensions, I'm assuming that this is with full planning permission, and not going to be done under "Permitted Development" rules?
Also, does the building HAVE to be that shape? It would be much better if it were just rectangular. Easier to build, faster, cheaper, better acoustics.... I'm assuming that it is that shape because of some type of restriction, or something on the land that can't be moved? I saw that you said " it maximises the full space available at the end of my garden", but I'm wondering of there might not be a better shape...
the height (3m) is the maximum I was allowed by the planning office, I was hoping for a bit more but hopefully it will be ok (will be around 2.6m acoustic height internally).
Height doesn't just go UP... it also goes DOWN. You can dig a deeper hole for your studio, go down some more, still have the roof peak at 3m, but have as much height as you want inside. Some of my clients in the UK have done exactly that... Of course, it does mean that you need steps down to get inside, and you have to deal with the draining the water tha runs down the stairs, but it's a really good option if you need the height. You show a baby grand, and you mention drums, so you most certainly DO need the height!
I struggled to work out if the proportions would be good as all of the room calculators I've found assume you have a rectangular room. I'm hoping the one angled wall will act in my favour in terms of room modes?
Calculators don't apply much in your case, because you have one steeply splayed wall, so you can't trust any number that is associated with that wall. The vertical and side-to-side axials will be fine, as will the tangentials that only involve the remaining five walls, but the obliques will all be wildly wrong, as will the length-wise axials, and any tangential that involves that rear wall. In other words, whatever ratio you came up with is not really valid.
And no, having splayed walls does not improve your modal situation: in fact, it usually makes it worse, because you lose support for one or more axial modes, and those are the ones at the lowest frequency.
However, don't despair! Room ratios and smooth modal response are nice if you can get them, but you have to treat them anyway, regardless, so it's probably not a big deal. As long as your modes are reasonably well spread out, not all bunched up around the same frequencies, then you are OK.
On the other hand, I don't see any provision at all for bass trapping in your studio, so it's going to be a modal mess!

You should work on that...
The maximum noise allowed in the UK is 34dba.
If I say a typical level for piano and light drums might be around 100dba,
Careful there! You are comparing apples to bananas! Not valid! The "A" weighting scale of sound pressure measurements is meant for low level noises (quiet) and speech, typically. ... because the "A" curve is a fairly reasonable approximation of how the human ear perceives quite sounds. The "C" scale is much more accurate for loud sounds, and music especially. ... because the "C" curve is a fairly reasonable approximation of how the human ear perceives LOUD sounds. But you cannot easily convert between the two scales, unless you know the spectrum of the sound. So normal speech will show up on both scales pretty much the same, regardless of whether it is loud or soft, but music is very different: there can easily be a difference of 20 dB or even 30 dB between measuring the exact same contemporary pop or rock song on the "A" scale or on the "C" scale. Not all decibels are equal! Some are more equal than others...

.
Fortunately, your regs measure in dBA, which does not take into account the low end. And you will be "playing music in dBC" inside, so you have a big advantage there. By "playing music in dBC" I mean that the music you play inside will be loud, and you will perceive it inside the studio similar to what the C scale measures: It will seem like dBC to you: loud bass notes will sound loud. But if the cops come along to measure, they will have their meter set to "A" weighting, so even if they can
HEAR the low notes, that won't show up on their meter! Because "A" is less sensitive to lows. In other words, you could be bashing out a bass riff at 100 dB, with pretty much all the energy in the low end, and you would measure that as 100 dBC. But if they came into the room and measured with dBA, they would only "see" that as maybe 80 dBA.
OK, so this is getting a bit complex, but it's important to understand the difference, when designing the isolation for your studio. And since you cannot compare dBA to dBC directly, you cannot say that if you have 40 dB of isolation, then the piano played at 100 dBC inside is going to be measured at 60 dBA outside....
You can't compare dBA to dBC.... Either you use dBA (which is no use for loud sounds), or you use dBC (which is no use for quiet sounds...)
So, in other words, you will need to decide how to measure! Should you use "A" or "C"? You might think that it would be better to measure everything in dBA, because that's what the regs say, and that's how the cops will measure if your neighbors complain... but you would be wrong! What matters, is dBC. Because that's what YOU will be hearing inside the room, and all of the sounds you will be producing are full-spectrum, and loud. It also means that if the cops do turn up and measure the level, they will see a LOWER reading than what you have, which is good for you...
So, from now on, use only dBC. Forget dBA, unless the cops really do pitch up! Do all of your measuring with your meter set to "C" weighting, and "Slow" response.
The nearest house is 25m away, so that should provide 28db reduction.
Perhaps... depending on what other buildings are around, as well as the temperature, prevailing wind, etc.
So I calculate that the reduction required by the building is 110 - 34 - 28 = 48db.
Assuming dBA....

A drum kit can easily put out 115 dBC, and more when played hard....
From what I've read I believe 48db reduction should be achievable with the materials / design I've chosen.
It's always better to shoot high, rather than low, for your isolation. How did you arrive at the conclusion that you will get 48 dB isolation from your structure? What calculations did you do?
The outer walls will be concrete blocks, the roof concrete beam and block.

Good start! That alone will get you decent isolation. How thick will the walls and ceiling be?
Timber frame -> 16mm OSB -> green glue -> 16mm plasterboard (drywall). Sitting on the same concrete base as outer leaf

good inner leaf, too. What size framing?
Inside out design: 16mm plasterboard -> green glue -> 16mm OSB -> 150mm insulation -> fabric. Built in 2.4m x 0.6m modules.
Make your modules smaller than that. Those would be extremely heavy to lift: complicated, and dangerous. It's better to have a larger number of less heavy modules. Easier to manage.
Floor
Wooden
Why? If you have a perfectly good concrete slab, then why would you put a wooden floor on top of that? And what type of floor are you talking about? You are on a VERY tight budget, so I would leave the floor out, and just use the slab. Nothing better, acoustically.
Window
One non-opening single glazed window in each leaf. 12mm acoustic laminated glass.
I can't see you getting 50 dB of isolation with such thin glass. That will be your weak point. I would suggest putting thicker glass in the outer-leaf at least. 20mm would be good. And the same on the inner leaf if you can afford it, or at least 16 mm.
Solid fire door in each leaf, with a door closer

... and multiple full-perimeter seals including threshold....
The silencer box
"The"? As in singular? Only one? You are looking for high isolation, in the region of 50 dB, so you will need two silencers on each duct: one where the duct passes through the inner leaf, and the other where it passes through the outer leaf.
is in the bottom of the bass trap (between the inner leaf and the outer leaf),
Sorry, I don't get that: How can your bass trap be in between the inner leaf and outer leaf? The bass trap is inside the ROOM, not inside the wall. And if you put an HVAC duct through your bass trap, then you are reducing the effectiveness of the trap! It's a small room, so it is going to need extensive bass trapping. You can't afford to lose efficiency.
the silencer for the outlet is on the outside of the building as I didn't think there would be room between the leaves on the other side of the room.
Ditto. See above. You need two silencers on that one as well, if you want high isolation.
I did consider putting the outlet in the other bass trap but wasn't sure if that would really encourage much airflow as they would both be at the same end of the room.
Right. Generally, I put the supply register at the back of the room, and the return register at the front, as far apart as possible.
I believe that flush mounted is the way to go,

Yes! Undoubtedly. Good move.
I've had to put them at 75 degrees instead of 60 otherwise the mixing position was too far back in the room.
That's probably fine, but do check your angles, distances, and probable acoustic response carefully, taking into account the actual speakers you plan to use.
I'm going to have a sub on the floor somewhere so monitors won't have to reproduce too much bass. I need to do a bit more research on the exact flush mounting assembly, but think I have a basic idea.
Sort of but not really!

For example, you have shelving at the point where the soffit meets the wall, with a huge reverse "kink" in the baffle. That is going to produce all kinds of messy stuff in your room, with major edge diffraction, reflections, delays, comb filtering, and other stuff. If you take a look around the forum at rooms with soffit-mounted speakers, you will see that it is NEVER done like that. For a good reason.... The speaker baffle needs to merge smoothly into the side wall, with as little disturbance as possible. I often put "wings" out to the sides of the soffit baffle itself for this very purpose. The wings help blend the soffit into the wall...
There's other issues too: your speakers appear to be in the middle of the baffle, which is not the best place. Offset to one side or the other, to reduce the "focusing" or "lobing" effect that you can potentially produce otherwise.
Storage
I've created a small storage area, with a light weight sliding door to avoid creating a 3rd leaf.
It's still a 3rd leaf, and the area is REALLY tiny! I don't see it being much use like that. It also takes up the exact location where you need massive bass trapping...
I guess the main thing is I'd appreciate any feedback on the overall design, a general sense check on things that I might not have considered.
Done! See above...
Is there any way of estimating if the room proportions are going to be reasonable given the angled wall?
As I mentioned above, you can estimate to a certain extent provided that you only consider the numbers associated with the other surfaces, ignoring everything associated with the rear wall. So you cannot trust any oblique, many tangentials will be off, and all of the axials associated with that wall.
Is it OK to have monitors at 75 degrees instead of 60?
Yes, provide that you understand what the consequences will be, in terms of sound stage, stereo image, and sweet spot. But it seems to be that if you re-design your soffits, you should be able to get a better layout.
Where is the intersection point for the axes of the speakers?
Could I put the outlet vent in the other bass trap in the other corner?
What size do your registers have to be? What is the flow rate that you calculated for your HVAC system? How much of that is recirculated air, and how much is fresh air make-up? What is the velocity that you are using at the registers? What duct size does that give you? How big do your silencer boxes therefore have to be? What is the total static pressure that this system creates? Is your fan able to handle that flow rate and that flow speed into that static pressure and with that duct size? What is the latent heat load of your room? What is the sensible heat load? What mini-split system have you found that has the correct capacity for dealing with that heat load, while also moving the correct amount of air at the correct speed?
Lots of questions you need to answer about your HVAC system! I get the impression that this is one area of your design where you haven't completed your research yet...
Also, your current plan has the air inlet down at the bottom front of your room, yet the mini-split is way up at the top rear of the room.... How do you plan to get the incoming untreated, moist, warm air, up to the mini-split so that it can be conditioned, cooled, and dehumidified? It somehow has to travel across the entire room, without mixing with any room air, to do that...

Not gonna happen! Think about that...
Should I go for 10mm glass in one of the windows and 12mm on the other? I'm not sure whether the isolation would improve because of the different resonant frequencies or be less because of the reduction in mass of one of the panes?
A couple of days ago I wrote a brief explanation for one of my clients who was thinking about the same issue, so I'll copy-and-past that here. Some of it might not be relevant (we were discussing PVB interlayers), but most is:
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OK, here's the "all you wanted to know about laminated glass" deal for today...
Acoustic PVB is a bit thicker than normal PVB, and with some manufacturers its also a special formulation. It greatly improves damping at the coincidence dip, and to a lesser extent, at other points. It also drives the dip UP the spectrum, since it makes the two layers of glass act more like individual layers, and less like a combined layer with higher mass.
Here's the difference:
laminated-glass--coincidence-dip--acoustic-pvb-vs-normal-NAMELESS.jpg
That's for two different types of laminated glass made by the same company. Same thickness. You can see that most of the difference is at the coincidence dip, but there is also an improvement lower down, until you get to about 200 Hz... However, below about 200 Hz, you are into another area, where there's some other type of internal resonance going on between the panes, that is reducing isolation slightly. So there are pros and cons. But mostly "pros"!
Now, about thicknesses: consider that the location of the coincidence dip on the spectrum depends on the density, but also on the thickness. If you are interested, the lowest frequency where coincidence can occur is:
Fc = c² / (1.8 * h * vl * sin²(a))
where:
c = the speed of sound in air (m/s),
h = the glass thickness (m),
vl = the longitudinal velocity of sound in the glass (m/s), and
a = the angle of incidence.
And of course, the "longitudinal velocity of sound" is a function of density (sound travels faster in more dense materials). For glass, it is about 5600 m/s (depending on glass type).
Thus, if you have two identical panes of glass, the coincidence dips of both of them will fall at the
same spot on the spectrum, and you will have less isolation in that frequency range. Any sound that gets through the dip in one pane will get through the dip in the other pane, because the dips line up.
By making one pane thicker than the other, you'd think that there must be an advantage, because the coincidence dip would also move, and therefore the curve would move. Yes it does! But not by enough to be hugely useful. Take a look at the curve again in the above graph for ordinary laminated glass: the dip is around 1700 Hz. Now, imagine if you could make the glass 10% thicker, in order to move the frequency down the scale a bit... Do the math: a 10% change in thickness produces only a small change in frequency (there's that nasty "c²" term on top....). It's a small change, but I'm too lazy to do the math right now, but lets' say, for argument sake, that you get a massive shift of 100 Hz by going from 12mm to 19mm glass. Imagine, if you will, that same curve above, and a copy of that curve moved over by 100 Hz to the left, so the dip is at 1600 Hz, instead of 1700 Hz. Compare the curve and it's copy. Look at the intersection of those two dips: Most of the dip is still in the same place! The majority of the "coincidence dip" still lines up between the two thicknesses of glass! You need a very large change in frequency to kill the overlap.
So, yes, there's an improvement, but it's not really worthwhile, because the dip is so wide anyway: you would need a very large change in frequency to get the situation where the dips no longer overlap at all, so you'd end up using one pane of glass that is really thin, and the other massively thick... The dip runs from maybe 1000 Hz to maybe 2200 Hz, so to get the dips to not align at all, you'd need to change the frequency of one dip by about 1000 Hz... good luck with that!
Plus we aren't even concerned too much about the coincidence dip anyway! It's way up near 2 kHz and even at the worst point, isolation is still over 30 dB! What is REALLY important, is how the glass behaves down in the
low end of the spectrum. The lowest point on the dip is over 30 dB, and that lines up roughly where the rest of the curve hits about 500 hz. Thus, below 500 Hz (the entire low end, plus half of the mid range...), isolation is far worse then even the worst part of the coincidence dip: It's that bottom ends that matter most. So the coincidence effect isn't such a big deal, really.
Basically: more mass is better. Thicker glass is better. And acoustic PVB is better than standard PVB. (With acoustic PVB, the coincidence dip stops being an issue at all.)
Bottom line: get the thickest glass you can afford, with acoustic PVB if you can. You want at least the same surface density as the rest of the leaf (but preferably higher, by a factor of about 1.4...

), and get the glass that has the best PVB (some manufactures use different materials, other than PVB for their interlayers: check the actual tested specs).
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Hopefully, there's enough in there to answer your question.
One other thing: your door doesn't seem big enough to be able to get a baby grand in and out of the room all the time... And also: where will you store the piano when you are using the room for other things? I can't see you getting a baby grand AND a large drum kit in there at once... plus, even if you could fit them physically, it would sound pretty bad...
- Stuart -