If I'm correct, it's not nuclear energy that blocked out the sun, but ash and some chemicals from the exploding volcanoes. Ergo no sun = frozen earth.
You don't seem to understand what Nuclear Winter means. Nuclear winter (also known as Atomic Winter) is a hypothetical climatic effect of nuclear war, or sometimes the effect of large amounts of smoke or soot is lunched into the stratosphere. Similar climatic effects can be caused by comets or an asteroid impact, also sometimes termed an impact winter, or by a supervolcano eruption, known as a volcanic winter.