In the show, I think the zombie plague has only been going on for a few months or so? I'm sure people aren't thinking of the nuclear reactors and more about trying to stay alive. Upkeep of all sorts of things would eventually create a whole lot of chaos... Pretty scary to think about.
I am being totally silly even posing that question, I know!
But I can't help it - when someone creates a sci-fi kind of world, I always wonder about stupid details like that.
You notice I didn't question the plausibility of zombie-ness itself? Because a virus that reactivates the brain stem is apparently something I have no trouble accepting, but those pesky nuclear reactors!
FYI, though - I believe most reactors would have their meltdown within 2-5 days after their cooling systems ceased to run. And there are enough of them that I think any human survivors might get a few months in a radioactive wasteland before they succumbed. It's part of the post-apocalyptic world that very few writers ever address. I guess I just like my sci-fi to be as plausible as possible.
Time for me to exercise a little willing suspension of disbelief, I think!