Sadly, getting an FAA licence issued on the back of your JAA licence will require a trip to the states.
In brief, the process is:
1) Pay £16 to the CAA for them to agree to talk to the FAA
2) Advise the FAA in Oklahoma City that you want an FAA ticket issued on the back of your JAR number
3) The FAA will now talk to the CAA and confirm your licence details
4) You will receive a note back from the FAA telling you that all is OK and asking you to nominate an FAA local office (FSDO) in the US at which you will appear to pick up your licence
5) You contact the FSDO concerned (they are all in the US), and turn up on the appointed date to receive your licence. No charge.
6) Before it can be used, you need an FAA Bienniel Flight Review. This is conducted with any US CFI or CFII and takes a couple of hours. A minimum of one on the ground and one in the air. The format varies from exam to normal-ish lesson.
That should just about do it.
2D