Cockroaches would not survive an nuclear fallout.

A cockroach’s ability to reproduce is compromised after exposure to 1,000 rads (a unit of radiation dose), which is the amount of radiation a cockroach would be exposed to if it were 15 miles away from the Hiroshima explosion. If all cockroaches were hit by 1,000 rads or more, the cockroach population would end pretty quickly. 6400 rads will kill most immature cockroaches, and 10,000 is enough to kill most adult cockroaches. That’s impressive compared to human beings, who can only survive 400-1,000 rads.

Cockroaches have nothing on the Habrobracun wasp, which can survive up to 180,000 rads, or the Deinococcus radiodurans bacteria, which can endure 1.5 million rads (3 million if frozen).
(source)