The way our heating is set up, we have a room thermostat that has a wireless connection to a receiver which in turn gives permission (or not) to a timer that then tells the boiler whether to fire up or not.
I thought it might have been a communication problem between the room thermostat and the receiver, but (after changing the batteries in the thermostat and rebooting the receiver, just to make sure) they did appear to be talking to each other.
My next thought was that there might have been an issue with the connection between the timer and the boiler. I traced the wiring to a junction box, inside which I found this:
I decided it was perhaps too dangerous to risk doing anything whatsoever to that.
Meanwhile, my wife had taken the controller casing off and found that it contained a circuit board. It was very loose, though, and there was a chance that the pins on the circuit board were not engaging with the slots in the housing that connected the controller to the timer. I therefore decided to put the circuit board back without the casing to make sure that the connection was good.
It's actually a lot harder to line up pins with holes than you might think.
Here's the first line of the email I sent to our regular heating engineer:
Our heating hasn't been coming on of late (the hot water is fine, just the heating isn't), and we figured it may be an issue with the thermostat. Whether it was or not is hard to tell as I accidentally blew up the receiver while trying to fix it.