I really do feel for you yelofox & if we were having the same problems with our neighbours, I'd write along these lines:
Dear Neighbour,
This is very difficult for me to write, as I don't wish to make waves, or for it to lead to a confrontation.
However, the loud music coming from your house in the early hours of the morning is disturbing my sleep pattern, bearing in mind that I have to get up for work at 05:30am.
I would really appreciate it if your music is either switched off at a reasonable time in the evening, or is turned down to a tolerable level, allowing me (&/or any other neighbours in close proximity) to sleep undisturbed.
Thank you for taking the time to read this letter & I look forward to a peaceful & friendly outcome.
Kind regards,
Mr & Mrs Smudge.
I would keep a copy of the letter & if they didn't want to play ball, I'd then contact the environment dept of the council.
Good luck.