Nuclear power is power (generally electrical) produced from controlled (i.e., non-explosive) nuclear fission reactions. Electric utility reactors heat water to produce steam, which is then used to generate electricity. In 2007, 14% of the world's electricity came from nuclear power, despite concerns about safety and radioactive waste management. More than 150 naval vessels using nuclear propulsion have been built.
In a Zombie Apocalypse scenario (or any kind of Apocalypse, for that matter), the inability to maintain the nuclear materials therein pose a considerable hazard for all life within distance of relevant prevailing winds of reactors in both power plants, and naval vessels.
Once a nuclear isotope has undergone fission, the core material will continuously produce both heat and radiation for a time and intensity directly proportionate to the material's radioactive half-life. In the modern day, this is carefully monitored with a variety of failsafes. Ultimately, those failsafes rely on its nation's economy and transportation system - either to bring the workers to the power plant, or to transport the oil, fuel, and maintenance parts that keep cooling system running. In a zombie apocalypse, the extensive maintenance a nuclear reactor requires will inevitably fall short as the economy, roads, and transit systems stagnate and decay.
A real life example of a narrowly avoided meltdown caused by inability to keep the oil based cooling system operational was the Fukushima Daiichi nuclear disaster of 2011.
Over time (a mere matter of days), any nuclear reactor will be overwhelmed by it's own heat, and a the plant will meltdown, releasing toxic radioactive smoke and debris into the atmosphere. This meltdown is not as destructive as a nuclear bomb (where the fission or fusion reaction unleashes megatons of force). The heat causes a large radioactive fire which rages on for days, and even after the embers burn out, the material will continue to emit radiation into the air and water of its environment, contaminating the area for hundreds, perhaps even thousands of years to come. As the winds blow this debris and cloud matter miles away, it will become an invisible killer of many survivors.
There are over 100 nuclear power plants in the United States, and hundreds more nuclear waste storage facilities that likewise require constant refrigeration.