Hi I'm so sorry to hear about all this. I've read this whole thread and laughed at lots of it, but having a nasty neighbour living next door is no laughing matter to live with.
I would be pretty careful about what you say and write back to them and anything that is said by either you to them or them to you, have it said in front of independent witnesses and any correspondence you might make about it to them, I would put cc to (appropriate authority i.e. Council, Community Police Officer, is there a Council based Neighbour liaison person for helping to settle domestic disputes? etc). That way you have verification of what is said and they can't say you are being unreasonable, swearing at them or threatening them etc etc.
I would suggest the first thing though is to get proper advice about what your rights are with the tree in your garden (yes I know it sounds crazy). Also find out what your neighbours' rights are about the tree in your garden - i.e. if they could claim it blocked their light, roots might be invading their property damaging fences or walls, you name it ask about it (find out what they could use against you should they go that route - after all he probably wouldn't be stupid enough to say to the Council or Police it was due to birds distracting him from the piano!). Forewarned means you are prepared.
Be sure about what a TPO actually means for you if you get one.
I would also ensure that you log everything that you are sent from them, keep everything. Take pictures of the tree in case anything does happen to it. If you are truely worried about what action he might take without your consent you could put a camera onto the tree which would show any 'work' he might do to it, but I wouldn't tell them that unless you mention you are putting a 'Birdcam' in for birdwatching. To be honest I'd probably not even tell them I was doing that. I'd also keep a log of all the piano practice -dates, times it starts, times it finishes, and whether you can hear your TV (check volume level on your TV) over it. I'd note beside them which piano practice sessions were difficult for you.
Did you say they've complained about your cat and their dog???? I'd note that down as well on any logs - date, what's been said etc. If they are starting about one thing they might likely start about others if they don't get their way. Andy sounds like the sort.
It might be worth getting the tree assessed by a tree surgeon when Andy/wife is not around, and maybe a report that its not blocking light, roots causing damage to their property etc, but I'd get that advice from the Council or whoever first off, as you don't want it to appear that you are thinking of taking it down.
I would also suggest do not lose your rag (I probably would as I'm short and stroppy but in this sort of instance it doesn't do you or your standing any good to be seen to be anything other than reasonable), so do not send any nasty notes or stroppy comments (even though he well deserves them), and do not swear (tempting I know) at him or touch him or his wife in any way (even a friendly touch on the arm to say can't we just get past this - you don't know what someone like that might say in the future - Watch Nightmare Neighbours/ Neighbours from Hell on TV).
Personally I'd put up the fat balls etc - its cheaper than a solicitor and take pictures of it being done. So you can be seen to be the one being reasonable. But that would be the only concession I'd make.
I am an older beginner on the piano myself, and have been playing for a year, and I can tell you it can get noisy (especially with the Tourettes I develop when practicing a new piece!)- luckily for us our piano is sited in a well insulated garden annexe on an outside wall but not with the sun directly on it - that's really bad for a piano - sun and even central heating or underfloor heating especially can really ruin a piano. So I don't have to be considerate of who might hear my practices, but if the piano were in our semi detached cottage, then yes I would. Its not reasonable to be doing that early in the morning or in the evening, or during primetime TV. I'd have to think about our neighbours and be considerate.
If you Google re pianos, you'll see that you can get electric keyboards but also something called Silent Pianos which I think are like normal acoustics but allow you to fit headphones to avoid the problem of annoying other people. He could also stand it on a thick rug to lessen the noise and I think you can also get something fitted into pianos to lessen the volume - I'd check that out (I might be wrong), my piano was a free one off of Freecycle, not the expensive instrument Andy's probably bought. I've seen pianos on the internet listed for anything from £2000 - £10,000 + secondhand (bang goes me upgrading!) so if I were spending that sort of money I'd want to make sure light and heat didn't ruin it. And I'd also google about soundproofing as if you were buying a piano and wanted to lessen the noise for the neighbours and what a neighbours' rights are re noise and pianos. i.e. be knowledgable so that YOU can be the one who is always seen to be reasonable and actually know what they are talking about in a calm sensible way.
I'd do all of that sooner rather than later, especially getting that advice.
What I can't understand is if he's really properly practicing why is he even noticing the birds? My piano is in shade, but its on a north wall next to patio doors looking out onto the garden and my chickens. When I'm practicing the only things I'm looking at is the music I'm trying to read and the piano keyboard if I'm getting the wrong notes. I don't notice what's going on outside because I'm not looking at it. And I certainly wouldn't be able to hear garden birds or my chickens clucking because the piano is loud enough I can't hear that and I'm concentrating. And yes if I was being distracted I'd shut the curtains!
Andy obviously isn't looking or concentrating on what he's playing!
So frankly I'd do what I suggested above and maybe invite him and his wife to some mediation with independent witnesses there, and personally I'd probably invite him to a local cafe or somewhere public to discuss it over a coffee. Listen to what he has to say, armed with what you know about your rights and his (as well as the environmental comments about mature oaks and wildlife alot of the other people here have said), and just firmly say you won't take the tree down, but have put up the fat balls etc his wife suggested as you are a good neighbour and reasonable person. I would tell him it was obviously his decision to learn to play which is great, but that putting the piano where he has is his decision. Obviously he's considered where he was siting it and surely took into account the view and any affect of light and heat has on a piano - he must have had proper advice from an experienced piano tuner before putting it in that position?!?! Therefore he decided to put it in that place and that is his choice and not a problem related to your property. You have tried to be considerate and help him, in the same way you will expect him to be considerate and not practice early in the morning, late at night or at a volume that will affect you and your family. Be sympathetic and encouraging him playing but let him know what you will not be happy to put up with and that you not only know your rights regarding the tree but his obligations to you regarding noise.
Then take it from there. I really hope for your sake he realises he's being a jerk and you won't be bullied and it all works out from there.
Will be thinking of you - Good Luck.