My understanding is that you will need to convert your licence to JAA, but I'm not sure if you can get a temporary licence to allow you to continue earning in the meantime. It is also (or certainly used to be) very expensive (£15,000 rings a bell).
Why don't you try contacting the 737 LCC's directly - Ryanair and Globespan in the UK - Easyjet are getting rid of theirs. Air Berlin in Germany also. They may be able to advise you on whether or not you could work for them. As I understand it there may be a possibility of working for Ryanair in particular with an FAA licence.