If I understood this correctly, the main challenge in nuclear bomb design is to time the conventional explosions in such a way that the resulting blast will compress the nuclear material, achieving critical mass, and not simply tearing it all apart.
So I think that this fire would probably not have caused a nuclear explosion.
Years of testing have never yielded a nuclear detonation from the conventional explosives cooking off in a warhead - radioactive strewn desert in Australia is a testament to the British testing this directly.
Nuclear bombs require absurd - something on the order of microseconds (I suspect nanoseconds probably come into play) detonation precision in order to reach super-criticality.
So I think that this fire would probably not have caused a nuclear explosion.