The fallout from the bombs irradiates the entire planet. It also causes a new ice age.
However, the rapid global climate destabilization caused by the war caused intense global cooling due to all of the nuclear fallout blocking out the sun.
EDIT: Oh no! Look two users above me. It's the alt account of PacBrad!
This ultimately causes society to implode. Wait, where did the girl go?
Easier said than done, especially since everything is now contaminated with radioactive fallout.