Firstly ensure that there is a demand for heating or hot water
Provided the boiler is getting the correct signals then it could be down to fan, aps, high limit stat, PCB.
If the fan light is staying off and the high limit light is staying off then i would say that it is more likely to be either the APS stuck in the NC position and not allowing the boiler to run the ignition sequence or the PCB is at fault and will not allow ignition sequence. The most common fault on this would be the PCB but if unsure then call out an engineer that will diagnose and fix for you.
There are quite a few engineers that will do a fixed price diagnostic repair and will only charge extra for parts should they be needed. We do this and there are a couple of guys on here that i know do this aswell.
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