During the Cold War, MAD stood for Mutually Assured Ddestruction. That is, both superpowers (mutual) could be certain (assured) that their arsenals would inflict widespread destruction on the enemy, even after a surprise first strike to disarm them by the enemy. They would have enough left for a second strike.
The arsenals today are much smaller than during the Cold War, but both sides should be confident that they can inflict unacceptable damage on the enemy (which is less than total destruction). For comparison, during the Cold War the United Kingdom had what was called the Moscow Criterion -- the ability to destroy a few major cities in the Soviet Union, in response to the Soviet Union destroying all of the UK. They thought that was unacceptable enough, and within their means. The US and Soviet Union had more overkill.
Russia seems afraid that Western developments in countermissiles and conventional (precision) counterforce weapons might endanger the Russian second strike capability. Or they just pretend to fear that, to justify their imperial aggression. My non-professional judgement is that MAD still exists this year, and for some years to come.
So imagine Russia taking out 150 big cities in the US, and the US taking out 150 big cities in Russia. There are more weapons than that, but they would fear missile defenses and put several weapons onto one target. Thee result would be bad, very bad, but not the end of mankind. (It is a good question of there would be targeting on cities, countervalue, or targeting on military bases which may or may not be close to cities, counterforce.)
It won't be the rich in bunkers who survive, by the way. It will be rural bumpkins far away (and upwind) from likely targets.
