To get a JAA type rating in North America, you must complete an approved course at an approved TRTO. The TRTOs approved by the UK CAA ar listed on their website along with the types that they offer. Interestingly, Active Aerospace is not listed so I would be very careful about accepting Panama Jack's advice. You cannot simply do a FAA (or TC) course and get a JAA checkride at the end unless it is approved by a JAA country
If you get a FAA (or TC) type rating, you will need to build 500 hours flight experience on the type before it can be transferred to a JAA licence. You must then pass a Proficiency Check on the type with a JAA authorised TRE.
If you have some operating experience on the type, but less than 500 hours, you will have to complete a course of training at an approved TRTO (but the CAA may allow a reduction from the full course to take account of your experience) and pass the written exam and Skill Test.
It's all in LASORS Section F9