Sure, but I’d bet that negligible time shaved off its lifecycle is less than the energy added by the nuclear explosion. The increased energy and outward force may even increase its lifespan.

That’s dependent on the nuclear pile going critical, which I assume it would, but actually working that out is… some ways beyond me, and highly depends on the missile design/explosion time/what you count as “increasing sun energy” versus “exploding next to the Sun”, I guess.