Mine recently stopped working, that is to say it works for a bit, then
stops, then works for a bit.

"Works for a bit" means that the clock is audibly ticking, and the
mechanical clock rotates. "Stops" means neither of these things happen.

To my untrained eye it seems the power is running dry on the clock. The
strange thing to me is that I thought boilers were plugged into the
mains, so how could it run out of power?

Now an admission: I made a post on this yesterday, but I think it was
over specific in terms of detailing the manufacturer (Sime) etc. What
I'm looking for now is just some general guidance on how these things
work, as I'll probably get someone out to do it, and I don't want to get
stung. I just spoke to a guy who was very vague, talking about some
piece of kit between the power and the clock, which could go wrong.

Thanks, Dan