I hold both FAA & JAA PPL(H) and did training on both sides. I'd agree with the post re Bristow being best for JAA in US. However, it depends what you are looking for. I did my initial FAA training training in FLA and when I flew into Bristow found it like flying into a hornets’ nest. My school had 3 helicopters, whereas Bristow had 22 in the air upon my arrival. It's very professional, but not if you are looking for a more relaxing environment where you can take breaks between flights at a picnic table under a palm tree. Several of the heli schools in West Palm Beach (such as Cloud9) are British owned and worth a look.
Personally, I'd suggest the FAA route for initial training and then come back to the UK and do an examination flight for your JAA.
Note to be careful that any part of the FAA training meets/exceeds the JAA requirements (such as differing x-country etc). Also, whereas the FAA also has the 5hr conversion training for differing types (ie going from R22 to R44), they do not have type ratings (not for smaller helis anyhow) and the JAA will only acknowledge the conversion training if done at a JAA accredited TRTO (Type Rating Training Organisation) for which most US schools will reply with “what’s that ?” if you ask.