You know those little rotating latches that they have on the doors of toilet stalls? In the UK (and elsewhere) they typically show “vacant” in green letters if the latch is unlocked, and “engaged” in red letters if the latch is locked.
So, yes, in BrE “engaged” is absolutely the standard and commonest term to describe a toilet cubicle that is occupied. (Though “occupied” would also be pretty common.)
And it’s also standard in the telephony context. That signal you hear when you try to call someone who’s already on the phone? That’s the “engaged” tone. And if you are reporting to someone else, you’ll say you couldn’t speak to so-and-so because the line, or the number, was engaged.