The birth of the new-type multi-mission destroyer, which is an ensemble of the state economy and an aggregate of self-reliant defence technology and which is equipped with the powerful cutting-edge strike weapon systems of the Korean style which do not allow comparison, is a brilliant fruition of the iron faith and will and outstanding leadership of the great Comrade Kim Jong Un, who indicated the path for rapidly developing the naval force of the DPRK into a powerful treasured sword for defending justice and peace and a service equipped with the comprehensive fighting power with nuclear weapons as the main axis and is dynamically steering the rejuvenation of the warship-building industry, and a valuable victory won by the strong aspiration and indomitable advance of the Korean people to outpace the world with the conviction in their own strength and courageous struggle.
http://kcna.kp/en/article/q/af24707e127494b9102a5ffa0145069f.kcmsf
idk how feasiblr it is to house nuclear-capable missiles on destroyers but they might be talking about missile defense, missile defense destroyers are an important part of other nations nuclear strategies.
A MiG jet can carry a kinzhal supersonic missile with a nuclear warhead. So I don't know why a big ol' ship can't.
you can put a nuclear warhead in an artillery piece, it's a question of range when the nuclear power you're dealing with is not next door. Being able to nuke SK is utterly assured but the DPRK is not trying to have nuclear deterrence against SK, they are deterring the US, whose missiles are either across the pacific or on submarines in that ocean--both prospects which can be guarded against with naval interception