All they need is a filtration system to filter the incoming air from radiation which is easy to setup.
All the Utilities on Earth will have been destroyed.
So, that is the water supply, and the gas supply and the power supply.
That means that in a bunker, you have the supplies and once they are gone, you have to go to the surface.
And If Nukes have destroyed everything "up there" then the fallout, the Radiation level will take a lot more time to dissipate, then those down below have supplies, like Water, Air, Food.
That Bunker is a coffin, nailed shut.
They're dead..
it'll just not happen as fast as it would have if they had been upstairs when the Heat from the blasts was similar to what God did to those Sodomites in Genesis 19.