Pastis,
JAA Modular route requires these bits:
1. ICAO PPL - So no need to convert FAA to JAA
2. JAA Class 1 Medical
3. 150 hours total time, 70 hours PIC (but get 100 for reasons I'll explain later) and pass 14 JAA ATPL Theory Exams. Hours and Exams can be done in any order.
4. JAA CPL can be done outside Europe - Test flight can be in a single or a twin, different schools offer it different ways.
5. ME if you didn't get it with CPL and can also be done outside Europe.
6. IR - 55 hour course required and must be done in Europe.
7. MCC course - Minimum 15 hours in a multi crew turboprop sim.
It is best to do all of the above at FTOs that are all approved by the same aviation authority. There are quite a few in Florida approved by the UK CAA. If you do bits with oversight from different countries' aviation authorities, you need approval to hodge podge them together and you'll often find the authorities don't like to grant approval or they make you do a lot of it over, especially the French.
JAA CPL issue requires 100 hours PIC, so get it during the hour building because you won't get much PIC during CPL/ME/IR training.