vortime
I can only give you an idea of what the FAA regs are. As bookworm sensibly suggests, if you are in the UK airspace you will be subject to UK jurisdiction. I am also not up to date with all these new US security rules which might impact on the issuance of an FAA rating.
I don't know of an FAR which prohibits FAA flight training (whatever that is if you are Part 61) in a G-reg. The FARs allow training performed by another ICAO instructor to be counted towards an FAA rating, but you need an FAA instructor to sign you off before you take the rating. When I did my FAA Instrument Rating, I logged actual IMC hours in a G-reg with a UK instructor in UK airspace, and this was allowed to be counted towards the required 40 hours of actual/simulated instrument time for my FAA IR.
The FAA also allows you to take an FAA rating in a G-reg if the examiner says it's OK. So the FAA appear to be quite 'easy' on training, or they did
If you know some FAA examiners in the UK, I bet there will be plenty of people wanting their details. There was/is one in Germany, I thought she was the only one in Europe.
Which is the 51st state? The joke goes that it's Canada, but Tony Blair does desparately seem to want that so called honour to go to the UK
bookworm, I don't know about UK regs, but your extract would suggest that all these FAA CFI guys doing FAA BFRs and IPCs in G-reg planes in UK airspace are being naughty.