Ice age and nuclear winter didn't occur, it all depends on the damage actually, Ice age would come from global warming, and nuke winter would come from nukes, but it all depends on how the weapon is used, nuclear weapon actually doesn't mean Atomic Bombing.
I'm more inclined to believe that whatever big weapons they made, they were in use over nukes themselves, so I am now correcting my opinion, but if it was a few thousands of years the nuclear winter could pass, as for Ice Age, that would last a longer while but after a few hundred years the climate would subside and balance out, which wouldn't explain the desert of the size of the Euro-Asia continents.
So yeah, more likely huge walkers.