Hi Dom,
I have a few other thoughts.
First, if the music is unhealthily loud and you are working with machinery which is loud, you may have a legal right to wear baffling ear muffs, to protect your hearing. This would also cut out the music.
When I read your question, my first thought was 'ear plugs?'.
Additionally, if the environment is so loud that you cannot carry on conversation with colleagues, you may be allowed to listen to radio through an earphone - and of course, if you could get ear muffs, this might something you could do without specifically asking permission.
Particularly if the management already have a licence to play radio music in your workplace (and it is not simply cds being blasted out over loudspeakers) this would be a way forward. But if they are not playing radio music, you may not be allowed to either, because it's a legal requirement to licence a place of work for radio.
It is definitely worth your ringing your local environmental health department for advice. You can usually be put through to them from the number for your town hall. They have instrumentation for measuring noise levels, and if the amount of noise caused by the machinery is above a certain pitch, there may be a requirement on Royal Mail to provide you with industrial ear protection. It may be an option you can take anyway, and they cannot prevent you, if you find the noise levels distressing...
For instance, if your ears are ringing when you leave work, that is a hearing hazzard. They cannot make you [u]not[/u] wear ear muffs, even if no-one else's ears are ringing.