The instructor needs to be JAA FI rated only if he is getting paid for it and the
training portion of the flight is within UK airspace.
The DfT permission is free to [part]-owners. I've got it many times.
TSA only applies to instruction by FAA instructors towards a specific FAA license/rating and only some (e.g. the CPL/CFI/CFII is exempt but the PPL/IR/ME aren't). "General experience" flights are not affected. A flight with a JAA instructor can be later used towards anything-FAA whatsoever - just because it just happens to count towards the FAA IR is just handy
and obviously TSA cannot apply there.
Most people are going to the USA for the FAA stuff, not due to training issues but due to the checkride issues over the years. It's a complex picture; e.g. if you want training and checkride in your own plane then you will have to do it here, but if you just rent spamcans then you can do it anywhere. But if you just rent spamcans why do the FAA IR? It makes sense only for owner pilots who go places.