You poor souls! Apparently you have the misfortune to live above people who refuse to be satisfied. You've gone to great effort and quite an amazing amount of money to accommodate these people. Send them a letter of notification via certified mail, return receipt requested (I apologize for not knowing the correct UK terminology, but this action will establish an official paper trail.) List all their complaints, by date and time, and the measures you've taken to remedy same, including your financial outlay. Lastly, state that further complaints will be considered as harrassment and will cause you to file appropriate charges with the authorities. This should take care of the issue. Good luck!
Kaye

About the shoes--is there a need to wear hard shoes in your bedroom, which is right above their bedroom? I do think they have a valid complaint there. Either keep the shoes off until you are out of the bedroom or put some sort of carpet/rug down. That should stop that complaint. I do think this is a simple thing you can do--it is miserable to be awakened by noise that can easily be prevented.

About the sax--for a year, I lived below a trumpet player. He also played a few other instruments. The most annoying thing was that he'd pick up the trumpet, play one song or some exercises, and stop. Only to pick the trumpet up an hour later and play a little more. Later in the day, he'd play it again.

Can your wife pick one time during the day when she will practice? Then you can tell the neighbors, "The sax will be played Monday through Friday from 9 am until 11 am," or whatever will work. Then they will know when the noise will be, and can plan around it.

For the rest, I agree they are nit-picking you to death with the complaints. I'm actually on your side about everything else. So what I'd do is just tell them you are through. Once. "We have soundproofed the apartment as much as possible. Please talk to your landlord if you think your unit needs more soundproofing. We will not be responding to any phone calls, texts or letters about the noise we make. We have done all that we can to mitigate that." Then follow through and ignore their complaints.

You might also contact their landlord and complain about them and their constant complaining.

My building was built in 1900. I have a downstairs neighbor who complains about the creaking floorboards in my bedroom. There is a fix for those, but the landlord needs to pay out the money for that--not me, I'm just a renter. But the downstairs neighbor complains every single time she sees me. I just smile and tell her to call the landlord and then walk away. I'm barefoot or in sock feet in the bedroom all the time. I can't help 100 year old creaking floorboards.

I agree with Kaye 100%, now its become bullying and they must stop, It sounds to me that you have gone out of your way to suit their needs, you have rights also. You might want to talk to a lawyer to find out your legal rights for your area. I hope you've saved all their text msg's for proof of harassment. Good luck
Christine

Thank you all for your responses! I really appreciate it.

I think @kaye820's suggestion of a written letter is a brilliant one and I'll start drafting one right away. We have kept all emails and text messages as @christine1950 mentions, so that will come in very handy for listing out, in plain language, how frequently they have complained and how acquiescent we've been.

As for @camlan's point about the hardwood floors - I've actually spoken to my wife about this and she agrees that although she feels it's her "right" to wear whatever footwear she wants in her own home, we will buy some carpet-runners to cover the walkable areas of the floor as this is a relatively easy fix. I will of course inform the neighbours of this.

I've decided that the main point of my letter will be along the lines of:

We *intend* to convert our loft so that the noise of the saxophone is greatly reduced for you. However, if you do not immediately cease your continual harassment, we will *not* go ahead with that plan. We have made every reasonable effort to respond to your complaints and will no longer be acknowledging any further text messages or emails. If you wish to contact us regarding any future noise issues, I expect it to be done properly, in writing or via your landlord, [name withheld]. I would also recommend that you also contact [landlord] to request that he looks into various sound-proofing solutions that could be used in your property.

Great letter, short & sweet and you've covered all the issues. I wish you the best, let us know how it goes.
Christine

Enough is enough, and I salute you for not taking this lying down, or standing up, or walking around in hard shoes. Here in the Colonies we have the expectation of reasonable enjoyment of our homes, and what this nitwit has foisted on you constitutes outright harassment. Should this situation continue or escalate, which may happen upon his receipt of your letter, do not hesitate to go to the authorities. And don't back down or feel intimidated by a bully who cannot be made happy. You are in the right, so go forth and enjoy your flat.
Kaye

Have you offered to come downstairs and listen to the noise (e.g. saxophone, or the cats trampling on the stairs.)?
Having lived below someone where I could hear every step and every cough, I kind of feel sympathetic. One feels at the total mercy of the upstairs neighbor, i.e. lack of control or being prepared. The suggestion to come up with a fixed time for saxophone practice sounds reasonable.
Of course, I very much sympathize with you, too.