I've checked and Knobby is indeed correct. The FARs aren't explicit in the section (61.75) on issuing the license on the basis of a foreign one, but section 61.56 says that "no person may act as pilit in command of anircraft unless" in the last 24 months they've completed a flight review. The FAA's FAQ on Part 61 of the FAR spells it out in an answer.
Chandler Air Service (seems like a popular place for UK PPLs) told me explicitly that I didn't need a BFR for the sake of the FAA, but it was a club requirement so had to have one anyway. Turns out they were wrong, but no harm done.