Again, the aliens can wait for the fallout to cool off. Especially if they use "cleaner" warheads and only high airbursts. Then they sit tight in their ship for a few years and wait.
The bigger problem isn't radiation, it's that humans are pretty hard to exterminate. Even in a full-scale nuclear war some people are expected to survive. And unlike many species humans live almost everywhere on the planet.
Not very hard to exterminate after an intensive bombing with nukes, asteroids, EMP weapons and other that reduced much of the city and technology to rubble and burnt ruins.
After that the only option for survivors is an all or nothing fierce defense of the territory like in stalingrad "We will defend the city or die in the attempt" and “Not a step back!”
will save us.