IO540 "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."
This may well be the case. However he does refer to training in the USA and hence my reply.
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 !
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.