I'm sure my JAA 737 IR won't cut it in the US!
Why not? It's an ICAO licence and the FAA is one of the most flexible regulating authorities around. Just because JAR isn't.....
You need to take a computer test which shows you understand the 'air law' of FAA instrument flight, called the Instrument Foreign Pilot test. You can even get the questions on the FAA web site so can have it all revised before you arrive. No flight test required.
You can take the test all over the place. You probably need to jump through hoops to get the CAA to send your details to the FAA before you arrive. Then I'd advise an IPC with an instructor, which you can include in your club check out and BFR.
FAA web site for a start. As for immigration requirements, I don't have a clue anymore, and I'd guess it is the same for your chosen flight school