My post apocalypse story takes place about sixty years after a nuclear war from the 1950s. Not only is the landscape bleak but the weather is as well. Where the temperature is cool and most days are overcast and you may get sunlight one or two times a week. Can plants, and in essence the planet, live on 1-2 days of sunlight? In my setting most of the old growth trees are dead from fallout and now there are young trees and plants starting to grow in their place.

I would say yes. There are plenty of trees that can grow in shade, and other forest plants under the canopy that flourish. They would take over when the plants that required direct sunlight were gone.