the idea is to nuke everything else so there’s nothing left for them to survive on.
Cockroaches are probably better able to survive the aftereffects of nuclear war than humans are. But they’ll still be killed by the bombs themselves.
and unless you nuke everything, they’ll just keep reproducing and coming back so long as there is something to eat.
If nothing else is available, they’ll eat each other. The sole survivor of Planet Earth will be one very large and very mean cockroach.