You can apply for an FAA license off the back of your JAA license. You pay the the CAA £38 to release your details to the FAA, you then go to FSDO in the US. Fill in some more forms and thats it. Obviously the process is a little more paperwork and time intensive, but this is the basics.
If you plan to hour build or fly in the states you will need to do this anyway, unless you only want to fly with student privelages which means, if it is to windy (12kts +) you can't fly, you cant take passengers, you are not allowed outside of a certain radius from the FTO...blah blah blah...
The best thing is the FAA don't charge a penny for the application or license issue...not even postage. the wort thing is the paper work and having to go to the states to attend at an FSDO to complete the application.