The above is correct in that you can hour build in any aeroplane and any country whatsoever.
If you decide to fly in America though you'll have to ensure you obtain an FAA PPL.
Quite a simple process of sending a fax to the FAA (in America) and one to the CAA (UK). They'll match the two up and the FAA will send you a letter of verification.
Once you have one of those you can head to America, go to the FSDO (local FAA office - one in every major town) and obtain your FAA PPL for free!!! Then a checkout with an instructor at the school of your choice to ensure you're safe to fly.
So go and get your JAA PPL at Cabair (or anywhere else) if you wish and then go and hour build in the USA!!! Don't limit yourself to JAA approved schools in the USA though - you can go to ANY school or club in the USA to rent an aeroplane.
No Visa is required if you're just going to be hourbuilding in the USA.
Best wishes,
Charlie Zulu.