Okay so I yanked off the lid of the cistern. After a flush, the water refills about half-way up the overflow pipe! Consequently, there's water leaking through a plugged hole in the cistern. This hole is where a toilet handle could be placed (it's on the opposite side).

Can't detect any leaks around the inlet/overflow pipes.

There's been a lot of water recently and part of it is probably due to condensation - it's improved since we've made sure the window is always open.

I've adjusted the screw on the balloon valve thing and made sure the water level is about an inch below the overflow pipe. We've flushed the toilet five times since and there's no leakage whatsoever.

Fingers crossed it stays like that.

Have you positively excluded tramps from the equation?