The rules have changed a bit since I converted to the FAA religion in '99, but I went into the FSDO office in Philadelphia with my "insert country" license and a medical and walked out 10 minutes later. Now you need to demonstrate Ameriglish capacity, but that's about it. No radio license.
BUT.
Beware, once you convert to FAA religion, all subsequent ratings MUST be done to FAA to count to your FAA rating. You can't get a JAA PPL, then convert religions (get your FAA PPL), then go back to your infidel JAA CPL/MEL, and come back to fly in the US of A using the higher ratings. It's a One-Way ticket baby.
'Nother But
It's not really a full-authority FAA PPL that you get anyway, it's a provisional based on your JAA rating. If the JAA rating ever expires (my "same country" PPL did just that since to keep that rating I must fly at least one hour every 12 months..... In "same country", with a "that country again" instructor. Wasn't aware of that detail, now was I?) your FAA ticket is as good as the paper it's printed on.
Talked to the FAA, they were kind enough to inform me that my hours count, but I'll have to do the written and flight test over.
If this isn't fun, why am I doing it?