I don't believe any of this is true—it certainly smells a lot like Cold War propaganda from the 50s. I don't believe we currently have anywhere near the nukes produced to have this happen even if we tried to perfectly cover the surface of the earth with nuclear explosions. To block out the sun we'd need something more like the volcanic winter of 536, continually erupting.

Nuclear war is bad, but it's certainly not "end life on earth" bad, or even "end humanity" bad by itself, just "deeply fracture humanity and probably enter a new era of civilization" bad.