Go to the CAA website, look for a document about approved Flight Training Organizations. This lists all the CAA approved FTOs (duh!). About five of them are in the US, mostly Florida. If you go to any of these organizations (avoid Florida during Hurricane Season though), you will receive a CAA/JAA compliant education and can even obtain a JAA/CAA PPL there (requires a minimum of three weeks though if you're ab-initio).
If you're learning to fly to the FAA syllabus (which obviously you can do anywhere in the US), I don't think you can count these hours towards a JAA PPL.
Yes, you're going to have to obtain a visa. An M-1, temporary student visa to be exact. Visa waiver only applies for holidays or temporary business visitors, not for students. Not a problem, just remember that in summertime a lot of university students apply for these visas too, and it might take a while to get an appointment at the embassy/consulate.
You also need a TSA clearance if you're an "alien" (that's the term they use!) and want to get flight training.
http://www.flightschoolcandidates.gov has all the info (if I remember the URL right)
The only establishment I know of in SW USA is Anglo-American Flyers. Never been there, no opinion. There's three places, I think, in Florida. I went to Orlando Flight Training. Search this forum for mine and other opinions. Also a lot of stuff on the others (OBE and Naples I believe) and Moncton in Canada(?).
Is it sensible - tough question. It is far cheaper than obtaining your PPL over here, mostly due to the lower fuel prices and the absence of landing fees. And the exchange rate right now is very advantageous. But you do need to have the money available in one go, you are going to be away for a number of weeks and if your aim is to obtain your PPL in three weeks, it's hard work - no time for other things. And you learn to fly in beautiful American weather (except for the occasional hurricane) and not in the crappy weather we have in Europe. OTOH - with the money you save you can easily buy ten additional flight hours with an instructor here in Europe to learn the differences.
Search pprune. Particularly in the professional study section there's a lot of threads dealing with all these issues, review of schools etc.