Scooter, an FAA Class three medical is only a Student Pilot License. It allows you to solo in US airspace after having been signed off by an instructor to do so, but it is NOT a PPL and it does NOT allow you to take passengers.
Bean_ian, I think, in general, you can count the three required "dual" hours for the NQ towards your PPL, but not the two solo hours. And I don't think that your restrictive JAA class 2 medical makes a lot of difference in this case. After all, all the dual hours you flew (day and night) were done on the instructors ticket, and all the solo hours you flew (day and night) were done on your FAA student pilot license & class 3 medical.
Technically, it might even be the case that you do not need a JAA class 2 medical at all to do your JAA PPL in Florida. You only need it once you start exercises the privileges of your JAA PPL in Europe. (But it would be silly not to obtain the JAA class 2 medical in advance. Just suppose you fail it for some reason...)