There have been manually controlled "criticality accidents" before, and I could see someone setting off a bomb by rope at a distance.
Would there be any prohibition involved doing so on Shabbos, as no electricity or fire is required?
One obvious issue is that there is bound to be some kind of life form in the vicinity that will die as a result, and this would be a clear "p'sik reisha" almost in its literal sense, with regards the melacha of "Shechita" (slaughtering).
On a day other than Shabbat it may well be "unnecessary" killing which is forbidden too (i.e. not being allowed to waste).
There are other likely "psik reisha" melachas formed, e.g. it will make a hole in the ground which is the melacha of ploughing.
(Disclaimer: I am not a posek. Please consult your Rabbi before firing any nuclear weapons.)
A single nuclear explosion could induce many Shabbat violations:
the energy of the nuclear bomb is actually not from nuclear forces but rather from the electrical forces of repulsion as Dr.Richard Feynman said
The energy which is liberated is the energy of the atomic bomb. This energy is usually called "nuclear" energy, but it is really "electrical" energy released when electrical forces have overcome the attractive nuclear forces.
this is essentially how a light bulb works for example. an electric potential causes charges to move and strike atoms releasing heat. the atomic bomb likewise causes charges to move due to the repulsion of positive and negative charges, releasing heat as explained by Dr.Feynman (see there for details)
I am far from a Rabbi so anything in this answer is just my thoughts
With that being said, I would imagine setting off a nuclear bomb (or any sort of bomb or weapon) at any time would be against Halacha. The only time where it would be ok would be to save someone's life (how this scenario plays out without harming other innocent people is hard to imagine, but technically possible) in which case it would overrule Chillul Shabbos and would be allowed anyway.