If you finish a JAA PPL in america, it will be in US (N-reg) aircraft. Hence the instructors will need to be FAA and hopefully JAA qualified. Once you do the JAA PPL skills test, as you have said you will need to post this off to the CAA.
In the mean time, your FAA qualified instructor will be able to sign you off as a student pilot and you will be able to hour build solo, but not allowed to carry passengers.
Once your JAA PPL comes through, you need to verify your license with the FAA which requires some paperwork sent to both CAA and FAA, and then a visit to a local FAA FSDO (Flight Standards District Office) to get a certificate issued to allow you to exercise the JAA PPL priveleges in an N-reg aircraft.
I had a rough experience with a JAA school in the states, if you want more info PM me, if you have the cash i would recommend staying in the uk. If an FAA PPL would be an option, I went to a cracking FAA school for hour building, an arrow for £57 per hour wet!