Also, I've seen the water coming through the neighbour's ceiling - and
it didn't look like it had come from the loo, if you see what I mean.

That means nothing. Often there is a blockage that then backs up. You need
to put quite a lot of water down before it backs up to the leak. Solid
material (for want of a better word) usually drops down the blockage and
only relatively fresh water comes out of the leak.

We also did lay off using the sink for the intervening days. So maybe
that's it? Maybe there's a joint output pipe for the sink and shower?

Definite possibility.

Christian.