In reply to Submit to Gravity:
First principles - check whether the boiler is actually receiving the call for heat! Room stat may be clicking but electrical contacts burnt out, a loose or broken wire might be preventing the signal being sent.
If you're comfortable with a multimeter you can check whether the voltage on the switched live into the boiler.
It could be a sticking diverter valve, if so you may be able to unstick the microswitch.
It's pretty easy to recognise, it's an aluminium cased rectangle above the pump.
It may also be a faulty flow switch keeping the DHW demand on even without a tap being on. That's a bit harder to recognise and can only really be diagnosed with a multimeter...