The great sleight-of-hand of the 20th century is that it never went away, we all just collectively decided to stop thinking about the fact that at every given moment for the rest of human history we'll never be more than thirty minutes away from total nuclear annihilation.
The world doesn't have enough nuclear weapons for "total nuclear annihilation".
The fall of the USSR chopped the legs out from under these stories. Nobody is worried about Iraq, Iran, or Afghanistan unleashing global thermonuclear war. That scare tactic just doesn't work. Even back in the 60s a lot of that paranoia was the result of internal propaganda.