Forty years ago, I read a report concerning the difficulties of protecting my companies equipment, which was to be installed in Air Force One, from a Nuclear EMP. An EMP generated by a high altitude nuclear detonation seemed particularly problematic. It raises concern about our power plants ability to safely shutdown and cool a core after a nuclear generated EMP. Adding to the concern is the fact that all civilian equipment would be inoperative and unable to assist.
Nuclear plants are designed with a high degree of passive safety measures, which come in far too many varieties to list here. Basically, the idea is that your systems need to be working correctly in order to maintain power generation. Normal operation involves actively suppressing physical processes that themselves suppress nuclear reactions. When something goes wrong, the reaction quenches itself.
For example, your fuel may be regulated by a neutron absorber whose cross section increases with increasing temperature, thus providing a negative feedback loop on the fuel temperature that necessitates no human intervention nor electronic equipment.
You should also note that the public perception of nuclear plants is way off-base when it comes to their risks. I've encountered many people in the US who legitimately believe that if you look at them the wrong way, most nuclear power plants will overheat and melt through to the core of the Earth (thanks to certain B-list science fiction films); they also believe Three Mile Island and Fukushima killed hundreds of people each. The fact is, the worst thing you can reasonably expect to happen to a nuclear plant not run by the disintegrating remains of the Soviet Union is that it will shut itself off and therefore fail to provide power. Besides, if there were a nuclear explosion close enough and powerful enough to fry all of your electronics, you have far bigger problems than an overheated nuclear plant.
An ElectroMagnetic Pulse (EMP) is generated high in the atmosphere as a hostile act in war, a nuclear detonation generating electromagnetic pulses at three levels, by the 1 MeV gammas of the detonation and their secondary effects, and by displacing the magnetic field of the earth and generating a pulse similar to the solar electromagnetic pulses.
Reactors are built with heavy shields so as to ensure that no radiation from the core escapes to the environment. This means they are also shielded for high energy gammas and secondary xray energies coming from above. In a war situation they should certainly be shielded with an appropriate metal shield for lower frequency electromagnetic pulses.
The only danger could come through surges in incoming power lines, from the area power grid, and if they are properly protected from surges there should be no problem of loss of control.