I believe that Nuke plants have on site back up electrical generation capability. Most likely diesel generators.
For an understanding of the vulnerability of our energy systems try reading "Brittle Power: Energy Strategy for National Security" by Amory and Hunter Lovins. The 1982 book was originally a Pentagon Study and rereleased following 9-11. The 500 page book can get a little dry, but I did read all 486 pages better than 20 years ago and it will scare you. More about the book here:
http://en.wikipedia.org/wiki/Brittle_Power