Using a separate computer normally results in your music on the phone being wiped due to FairPlay DRM when syncing. You can set “manually manage music and content” in iTunes for the device to allow copying to the device
There’s been no DRM on music bought through iTunes since April 2010.
Seems to be a bit of confusion, itunes doesn’t show the phone as a ‘drive’, it just shows it listed as a device, same as an iPod or pad. If the tracks are already on the UK computer, installing iTunes and selecting ‘manually manage music and video’ will avoid any embarrassment like the entire phone library being replaced.
Depending on version, the phone should show in a list on the left hand side, and if it’s the latest version, it should show up under the little device icon along the header bar at top. Clicking should show the phone and the ‘manually manage…’ option.
Drag the tracks from wherever they are into iTunes, then you can drag’n’drop in the latest version, I did it last week with an entire album, or, of you hover the cursor over each track, you’ll see a little circle with three dots in at the end of the track title; click, and you’ll get a drop-down with the ‘Add to:’ option, which should have the phone listed, then just click on the phone and the track should copy straight over.
Hope this helps a bit.