I would think that a nuke plant could generate its own cooling power with no reliance on the grid. I don't know much about nuke plants, but I would design that into them.
I have a friend who is active on this issue, a founding member of an organization which is actively lobbying and harassing congress. I know some of the inside story from him.
As I understand it, a nuke plant can not generate its own power if the grid is down. In fact all power plants draw power from the grid and one chicken-and-egg problem of the grid going down is you can't restart a power plant without grid power - a so-called black start. Obviously it would have to be done but it's not easy.
With the grid down a nuclear power plant has to rely on backup generators to provide power to the pumps that circulate water through the core and through the spent fuel cooling ponds. The cooling ponds are a bigger problem than the core itself as there is so much more radioactive material there. Without a continuous flow of cooling water, the water in the spent fuel ponds will boil and evaporate and the spent fuel will eventually catch fire.
Also as I understand it, a nuclear plant is required to keep a minimum of 30 days of fuel for the backup generators on site. Most plants keep only this minimum required quantity.
So in a grid down situation, one of the priorities would be to continue to supply diesel fuel to nuclear power plants beyond the 30 days on site storage. If the general situation deteriorated so that fuel could not be supplied then the spent fuel ponds would burn.
If you're serious about surviving the apocalypse you need to make sure you're not downwind - i.e. East - of a nuke.
