training in the US, clocking up the 1,500 hours and literally having an unfrozen ATPL as it were in the States.
Basically that's correct. BUT, the composition of this 1500 hr requirement for an ATPL between FAA (US) and JAA (EU) is different!
In the US the 1500 hr can be all in small, single-pilot aircraft (like Cessna, Piper etc.)! Not in Europe though, according to JAR-FCL at least 500 hr must be in multi-crew airplanes (either JAR25, Transport Category, or JAR23, Commuter Category).