Our toilet cistern works perfectly well, until either a bath is run, or the tap in the kitchen is run, then the cistern overfills and overflows into the bowl. What on earth can be causing this?

It should have pipes ( or tubes) that run from it, into a drainage field. The water has to go somewhere. They may be blocked or damaged.

How old is your house and what material are the pipes? Copper, galvanized, or plastic? When you open more than one faucet does that reduce the stream? Back to you, Tom