Can an atomic bomb/rocket with nuclear warhead's nuclear explosion be triggered by a conventional explosion close to it?

a thermo-nuclear explosion with the destruction pattern one would expect from an atomic bomb

This isn't a binary disjunctive, with purely conventional explosions on one side and the full-blown design yield of the nuclear weapon on the other.

Instead, there is a continuous spectrum in between, which goes through 'dirty' bombs (i.e. conventional explosion which disperses radioactive material) and partial nuclear explosions, where the core achieves criticality and contributes to the explosion (possibly including a proper fireball) but without the full destructive power that can be achieved in an intentional detonation.

Nuclear weapons design includes taking great steps towards avoiding accidental nuclear detonations, and particularly towards making it impossible (or as hard as possible) for any part of the core to achieve criticality unless the bomb is armed. As such, it is essentially impossible for the weapon to achieve its full design yield (or higher*) in anything other than an intentional detonation. It's also basically impossible to achieve any criticality in the secondary fusion explosive through any accidental explosions $-$ the steps required to achieve this are just too complicated.

However, that said, it is definitely not impossible that an accidental explosion could induce some level of criticality, leading to at least a small fireball and significant devastation.

For an accessible introduction, I'd recommend Scott Manley's Going Nuclear series.

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*Keep in mind that in most modern nuclear weapons, the effort goes into making the yield smaller, not bigger.