Although a nuke blast is bad, especially for the folks right near it, it’s still quite limited. It’s definitely not a globally catastrophic thing. Chernobyl was orders of magnitude worse caused by a burning nuclear reactor on fire for days/weeks. But that still only contaminated a swath of Scandinavia. Let me reiterate that this is still way way bad. Even a ground burst nuke would keep the worst of it right there. If they somehow lobbed one on Alaska, pray for those in Anchorage, but the rest of North America would have barely detectable radiation.
The effects you are describing are more with a WWIII with Russia where they launch thousands of missiles at every major city which would then burn as huge radioactive craters for years. That is Mutually Assured Destruction…
