"A U.S CFI cannot legally train you without such approvals/visa in place."
This one has been done to death here, but...
A US CFI can train outside the USA. A visa is obviously not required. TSA approval for the instructor would be if he's training for the PPL, ME or IR, not for CPL, CFI, CFII.
However, the FAA accepts all non-FAA flight training towards any FAA license/rating. So you could do all training you need (for the FAA) in the UK, say a G-reg, with a JAA instructor, and this avoids all the TSA stuff because it's nothing to do with the FAA. You just conveniently happen to do some flying which meets the FAA requirements......
Finally, there is nothing wrong with being ambitious!! GA is full of people who tell you this and that can't be done. Most of them are good at propping up the bar, and have never been beyond the local burger run. Also, JAA (EASA) is not the only jurisdiction under the sun; in fact most jets operate under one of the others.
You don't need an ATPL to fly a single pilot jet.