Just been down the cellar in a friends house to get a meter reading, and
when I touched the meter, the board fell away from the wall.

I have temporarily secured the board in place by tying it up with cord (to
prevent any strain on the cables). Is the refixing of the board the
responsibility of the house owner or the electicity company?

It appears that the board came away due to dampness in the cellar causing
the fixing screws to rust. I suspect that there are regulations regarding
electrical installations and dampness, is it likely that the electricity
company would refuse to reinstall (if their responsibility) due to the
dampness causing the initial problem?