If humankind were to come to an end as a result of a massive global nuclear war, cockroaches could plausibly survive the aftermath.
Although a cockroach within the blast radius of any nuclear explosion would be incinerated just like anything else, roaches would be better able to cope with the fallout as they are able to tolerate much higher levels of radiation than humans.
Cockroaches have a greater tolerance for radiation exposure due to cycles of the roach's shedding its skin. When cells divide, they are at their most vulnerable to radiation. While human have cells that are dividing constantly, roaches molt about once a week at most, which makes radiation's window of opportunity to attack cells much narrower, as noted in Discovery's Mythbusters database.
The cockroach's ability to withstand radiation isn't unique in the insect world or even all that impressive by those standards. Others, such as fruit flies or Habrobracun wasps, can endure much higher levels of radiation.