Julian...you saved me a lot of typing.
Bere...do as he/she says...but consider taking R/T in UK BEFORE travel to USA...it's much easier to learn it correctly from the the start - and the phraseology will be acceptable to the USA....learning the appalling rubbish that they speak out here...which is NOT acceptable in Europe...and then trying to learn it correctly later is much more difficult.
JLC - no argument from me...but I know of THREE students that have applied to UK CAA with JAA medicals and have been refused in the last few months. Word from Battleship House was that they must do EVERYTHING under the one Country...including medical (this appeared to change within a week...sudden refusals!!).
Sensible...yep...FAA "Student Pilot Certificate" is all that the LAWS require you to have in FAA airspace...and a UK one is invalid as a student. A UK/European medical IS VALID as a licensed pilot...not as a student....however, the JAA course syllabus suggests you should have a JAA medical as a prerequisite PRIOR to commencing flight training. (Paperwork thing....not law).