a) instruct and do BFRs (FAA) in a N-Reg a/c ?
b) instruct and do BFRs (FAA) in a G (or other JAA) reg a/c ?
An FAA CFI can do BFRs in any aircraft reg.
However:
Under the UK ANO, Article 36, any instruction in UK airspace requires a JAA instructor rating - unless the instructor doesn't get paid - regardless of aircraft reg. There's also 26(4)(a)(ii) which applies to the use of foreign licences in G-reg planes. Any instructor getting paid in an N-reg also falls foul of Article 140 (this applies to a pure JAA FI too; no reason why you could not do your whole JAA PPL training in an N-reg otherwise).
My understanding is that this means that:
An FAA CFI can do a BFR in an N-reg if he doesn't charge for it (or, if he does charge, does it in airspace outside the UK ANO jurisdiction*). The student is normally PIC but the instructor can be PIC instead.
An FAA CFI can do a BFR in an N-reg if he does charge for it but then he also needs a JAA FI rating (which nowadays means sitting the CPL exams, etc, as discussed in another thread) in order to legally receive the money, and the aircraft owner needs DfT permission under Article 140. The student is normally PIC but the instructor can be PIC instead.
An FAA CFI can do a BFR in a G-reg but he also needs the JAA FI rating regardless of whether he charges for it. The instructor is always PIC on any training flight in a G-reg. This is true in any airspace, UK or foreign.
In all the above, a BFR is just the same as any other sort of flight training.
Practically, you can get DfT permission (Article 140) for flight traning in a foreign reg plane, subject to the ownership rules specified on the DfT website. Basically this means you have to a part owner, with under 5 owners in total. Then you can get a BFR with an FAA CFI, but he still needs the JAA FI rating if he takes money from you. Unless he does it outside the UK airspace*
* this obviously means abroad, but the Channel Islands and the Isle of Man are interesting possibilities to look at - they have their own ANOs.
It's quite a mess. In practice, the charging issue (i.e. avoiding the need for a JAA FI rating, and/or avoiding the need for the DfT permission if say there are 5+ owners) is done by flying (with the instructor as a mere passenger) outside the UK FIR and then doing the training. Or (rather less legally) by a charge for "ground school".... just like training in Private CofA G-reg planes has for decades been done in the UK.....
I think the above is right. Would appreciate references if not.