I am sure you would agree it is a rather tall order to obtain training in the U.K for an FAA licence by a non FAA instructor and to then have the 8710 signed off by an actual FAA instructor (who I imagine would not be willing to do such a thing having not provided any training).... And all within a short space of time !
OK, the way you play this particular case is you obtain the training anywhere in ICAO-land, and then you fly the last 3hrs in the 60 days before the checkride with an FAA CFI/CFII. This FAA instructor needs TSA approval if applicable.
One could just get signed off by an FAA instructor but this is most unlikely since nobody will sign you off as ready for the checkride unless they actually flew with you.
Remember that even a U.K based FAA instructor must obtain TSA approval to give ANY training for the initial certificate issue. Ultimately it would be very difficult to get around the TSA requirements for a non resident/non US citizen wishing to gain an initial FAA certificate.
Not quite true. If the training is clearly towards an ab initio PPL then I agree. If the training is ambiguous and is say a 300nm VFR dual x/c (which is a CPL requirement) then no. Most flights you can do with an instructor are ambiguous in this respect; it is the license/rating which that logbook entry is subsequently used for (perhaps years later) that matters, but nobody will question whether some flight you have in your logbook from 3 years before was done under TSA. A BFR or IPC is also TSA exempt, and you can do any kind of flying within that.
I agree with MM Flynn about the light jets. There will be ways to fly them SP without doing a full ATPL. Not everybody likes this; during a presentation in West Drayton it was claimed that the CAA will ban SP jet ops in UK airspace, and the Germans were going to ban all jet ops without an ATPL. Neither "proposal" is likely to get anywhere as it would be a blatent breach of ICAO, and anyway what about all the SP jets flying already on a CPL/IR? It is also in the insurers' interests to get people flying, but flying safely. A couple of years ago I was offered a CJ1 with reasonable insurance (I think it was a CJ1 - don't particularly keep an eye on jets) if I had 800hrs P1, an IR, and did the TR at some specified establishment.