This is largely due to the fact that they have an incredibly solid exoskeleton, and they live happily on the most disgusting food.
Here is a plan to help in case such a holocaust occurs. Build yourself a full-body suit out of cockroach carapaces, and train yourself to eat the most disgusting foods. You may lose some friends, but they won't be around afterwards anyway. After the holocaust, expect to be considered King by the other cockroaches. There is a slim chance that they will have gained some super powers from the nuclear event, and will decide to exact a horrific revenge on you for your massacre of their friends and family for your own selfish ends. But that's a slim chance, so it's got to be worth a try.