Well, despite my very low expectations, the guys (specifically Viktor from Sweden and his men) turned up on Tuesday to re-level the floor. A huge truck parked outside and offloaded the rig (which just looked like a shipping container to me), then they ran all their special hoses up to the house.
They drilled lots of little holes (hoping to avoid the underfloor heating coils)..
Then started injecting their special goop which spreads under the floor, gently lifting it as it goes.
There was a bit of a 'moment' when the tiles all seemed to start cracking:
But with a bit of fiddling, they managed to squish them back together so you can't notice:
Floors are a bit tricky to photograph, when it comes to levelling, but here is a before (it was down by 26mm in the corner there):
...and after:
Before in the living room (again, about 25mm):
...and after:
See? It's kind of hard to appreciate, but we notice it immediately when we walk in the room now.
And, the best bit is that the underfloor heating is still working....
Only one gripe - they were supposed to fill in the drill holes with grout, but were in a bit of a rush so missed some.