Without photos or schematic it's difficult to offer specific information, but I can hopefully offer a few tips...
- The ignition transformer is normally a high turns ratio step-up wound on a ferrite rod with a lot of insulation (potted). It's not intended to give continuous spark, just 'ticks'. It's driven by capacitor discharge (via the SCR) rather than AC. The turns ratio is very high, so the primary will just be a few turns. It's DC resistance will be very low. It's very unlikely that the primary would go short (as long as you don't start sticking higher DC current through it

) Insulation breakdown of the secondary would be more likely, but I can't see that blowing the fuse, you just wouldn't get a spark. Check that the SCR isn't shorted, but apart from that, it probably isn't a particular suspect.
- If it's blowing the 12V fuse, then I'd start at the input. You ought to be able to measure a short and track it down. If the supply is 12V AC, then check for so shorted rectifiers etc. Strange that you haven't located anything like this - try drawing out a partial schematic to understand it a bit more. Do you actually mean 12V? These boards are normally directly mains powered. [Edit: I just re-read your post. If it isn't drawing excessive current on it's own, then it might be an external wiring fault or part].
- The normal failure modes on these 'sort' of boards are dried out electrolytic caps and bad solder joints. It's unlikely the either of these would cause the fuse to blow, you'd just get false trips, screwed up timings etc.
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A note of caution: Messing with any internal part of a gas appliance without being Gas-Safe (?) registered is illegal in the UK. If you screw it up, you could do something like accidentally turning on the burner control valve and leaving it that way. That could cause a nasty explosion if the gas builds up and then gets ignited. Please consider yourself advised.