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Modern reactors in the U.S. have "failsafe" shutdown. If power is lost the control rods drop (gravity) and stop the reaction.
Nuclear power keeps getting safer, but the older plants especially in poor countries like much of the former soviet union (like chernobyl) are the ones to be VERY concerned about.
I would not really worry over-much about internal problems at a nuclear powerplant. Despite popular fears they can't blow up, and in the US at least have enough built in safety systems to prevent any serious environmental problems.That makes me feel a lot better.
Unless the event instantly killed off a large number of the population, there would still be workers at the plants for at least a short time. I can't imagine that they would all just flee without shutting down. These guys would know better than anyone the consequences of abandoning an operating nuclear power plant. I would also hope that they have contingency plans for this kind of situation.