This video ignores the mechanisms and decision-making processes people use to prevent nuclear escalation. And as far as accidents go, it takes a lot to make a nuclear device go op in the proper manner. If an external explosive were to hit it, it'd probably just cause a localized "dirty" explosion rather than the full nuke going off. In fusion devices at least, the purpose of the internal explosive package is to create the kinds of pressure needed to get the fuseable material to fuse properly. Within a fission bomb it's more likely, but you'd still need the explosion to hit at juuuuust the right point with juuuuust enough force to dislodge the neutron pellet without blowing the tamper apart (while U238 is dense, a missile can cause serious damage to it.)
In most fission bomb designs, a missile striking the bomb would cause a hell of a radioactive cleanup, but it wouldn't necessarily cause a chain reaction. Then again, the video and I presume the book rely on the layman being, well, a layman. Ignoramuses eat this shit up, I guess?