The basic position of the pilot license country matching the one of the aircraft reg is what gives you worldwide (ICAO) privileges.
In practice, the is true only for private VFR flight. Anything commercial (and that includes offering flight training) is locally regulated, for domestic trade protection reasons.
There are also various concessions. The CAA automatically validates ICAO licenses so you can fly a G-reg, worldwide, on any ICAO PPL, VFR, without having to do anything. I am sure there are other such concessions but haven't bothered to research the subject.
One can go beyond this, with extra work (which might be just paperwork). One can get a piggyback license. Or with some paperwork and a little bit of flying and a checkride, one can get a whole standalone foreign license, making full use of one's previous UK PPL training.
Ratings are something else. To get worldwide IR privileges, the two countries must indeed match. I can think of one exception (Greece and FAA IR) but that's no use to anybody in the UK.