Does anyone know where I can get the Unreal Gold and Unreal Tournament soundtracks in MP3? Or alternatively how to convert the Unreal .umx music files into MP3?

Modplug player can play umx files and save them as wav files.
From there, it shouldn't be too hard to find something to convert them to mp3.

Thanks, I'll give it a try.
Excellent. It plays the umx files without any problems on my machine. I did wonder whether Vista 64 might have issues running it but nope, all was fine. Better still it's just an extract from the zip file, no installation needed, and as far as I can tell, no registry entries.
Many thanks ZamFear. +1 rep.

foobar2000 has the foo_dumb plugin which allows it to play modules. From there it is also trivial to convert the files to any format you want.

This, or Winamp's function as well. I used Winamp. Just Chipamp (OCRemix's project) and output them to a WAV file (you can change this in the settings). Use that to make a nice sounding CD, or compress them into MP3 files. The same can be said for using foobar2000, if my memory serves me correctly.
I wish GOG.com would include those soundtracks; they're amazing.
EDIT: Ahh, nevermind, the original suggestion is a bit simpler. Never mind. :)

Well, here's 3 1/2 hour version of the full soundtrack on YT (that's usually what I link to ;):

http://www.youtube.com/watch?v=9IISjIMpepo

Oh, and of course there are single songs available, starting here (for one of the best):

http://www.youtube.com/watch?v=6YWfb3ti4V8

Modplug isn't ideal if you want to do large batches. It's otherwise ok, since it supports UMX input and WAV output out of the box, but like you noticed, it can only do one at a time.

If you want to do an whole playlist's worth at once, you'll need to do like other people have suggested and use something like WinAmp or Foobar2000. It just takes a bit more work is all. You will need to install input plugins that can handle UMX files, and then perform whatever rain-dance the player requires to output to file instead of audio.

I found I have this WinAmp plugin installed, but I seem to recall installation required some manual intervention (if I remember right, I had to correct a line in the section it added to Winamp.ini. The line "in_mod=" is supposed to point to the input plugin for MOD type files. For most people this means it should be "in_mod=in_mod.dll" but by default the installer sets it to something else). But once you get past that, it does work. There's also one here, but I haven't tried it and don't know if it actually works properly. From there, it's just a matter of going into WinAmp preferences, and switching the output plugin to "Nullsoft Disk Writer." Check the disk writer plugin's configuration to make sure the output directory is where you want, and hit the play button.

As Gragt mentioned, Foobar2000 should be able to handle this as well, but I've never used it so I can only give limited guidance. It looks like this is the appropriate input plugin and basic instructions on how to convert are found here.