There are certainly advantages to doing as much of your training in the US as possible... not only much cheaper..especially with the dollar now regarded as almost a third world currency...but also in places like Florida and the west Coast the weather being so much friendlier means you can get the thing done a lot quicker... the more you fly the easier the whole thing comes together.. many in the UK struggle to find the money for the hours and then struggle to find the weather to fly.. this means often that the first half hour of any lesson is trying to catch up on what's been forgotton since flying maybe 3 weeks ago... bit like ski-ing really
There are a number of JAA qualified schools now in the States... looking through the mags will suggest a few.. as for their reputation... try the wanabee forum on PPrune.. or any other forum you think might suit...the Mods will move the post to where they think it should be...
One rule applies to any training organisation no matter whether it's FlightSafety or Arnies Flite-Club... NEVER PAY LARGE AMOUNTS UP FRONT !
Pay them enough in advance to see you through the next week... if they're not happy with that then pay someone else.
As for getting both JAA and FAA.... do that only out of necessity... as I did... remember... the FAA is still pretty useless outside of the US and in the more remote areas of the world... That doesn't include Scotland
