The effect of Nuclear Fallout comes up a lot the immediate survival post-apocalyptic type questions.
In these situations the power grids are all off, and there is usually a drastic reduction in the human population - we can expect that there will not be many nuclear specialists left around to maintain the power plants.
My research indicates that:
Nuclear powerplants storing spent fuel rods outside of the core will explode if they cannot be kept sufficiently cool. This cool documentary says this will probably happen about a week after the apocalyptic event.
http://topdocumentaryfilms.com/aftermath-population-zero/
There are many hundreds of Nuclear Powerplants in the Northern Hemisphere especially North America, Europe and Japan.

All these explosions will cause a lot of smoke containing highly radioactive fission products. This smoke will then circle the entire Northern Hemisphere for a few weeks, and then precipitate out in the rain.
Things would would not be as bad in the Southern Hemisphere because there are a lot less nuclear powerplants, and apparently the winds don't really cross the equator.
Questions
What is the impact of losing power/operators to all the nuclear power plants in the world?
Will the Nuclear Powerplants definitely all fail?
Should all future post apocalyptic movies/tv shows be set in Australia because people in the Northern Hemisphere will be dead?