Which locality (if any) uses "at the roundabout, take the second exit"

This is exactly what is used in the UK. It's in the Highway Code.

It has the advantage that it's completely unambiguous, and it's easy to count exits as you're going round, particularly on large roundabouts at motorway junctions and the like. My car's satnav combines the two ("At the roundabout, turn slightly right: take the second exit.")

I suspect that Google has paired the British "roundabout" with the British numbered-exit route description. You've inherited roundabouts and Google delivers them with other colonialisms.

The US has "traffic circles", I believe; I've no idea how they determine how to navigate them, but it's likely that Google would pair the US route description with the US name for roundabouts.