Sir, I think It dose not work that way.
TLDR: The reactor cores will be safe no matter what. Until someone genuinely screws up.
Full format: Those power plant will be safeguarded against any such events. Plus a national grid may trip not the reactor. One should understand that a Nuclear Power plan like hydro has two component, the power generation and the nuclear or the hydro part (pls accept this tongue and cheek nomenclature for explaining in a highly abstracted scenario. ) When the grid trips the power plants feeding it will be the first to know it. They can reduce the load of the power plant to suite the need. They do this either by dis-engaging the turbines from the steam or just closing the gates for dams.
I will clear one thing, Grid Shutdown dose not equate Reactor Shutdown. Sometimes power generation can be the byproduct of it's primary task. They can shutdown the power generation without even messing up with the core. And if it's known that the grid is down for a good number of days then the management will be more than happy to shutdown the core. Why? Bcoz a tons of maintenance activity could be done with, which would otherwise would have to wait till the next fuel cycle. Plus rigorous safety check can also take place.( Those with safety inspectors in NR are some harda$$es !! )
Unlike popular belief, emergency shutting down a nuclear plant, if the need arise to, is an automated process where humans have no role to play, due to self preservation. If some critical system goes out, the backup kicks in. But if all such back up fails, which is highly unlikely as backups are quadrupled even for non-essential systems, the mechanism to shutdown kicks in. The automated system has it's own power supply, both on-board and streamed from remote sites. The on-board system consists of multiple small gens with their own fuel reserves. These systems are untouchable by any such "Fuel Shortages". Moreover until great mechanical failures occur, the plant can generate enough power to sustain itself. Unless every goddamn system fails, meltdown won't be possible even if someone wanted to. Even then the control rods can be "Gravity Dropped" into the core, to cool it down.
So for such a minor grid trip, like this one, the nuclear power plant are safe. I will go out on a limb to say that they are probably the most secure places to be in during such stuff. There is always an outlet around to charge your phone ( to play awesome music) and rail-gun (to ward off those waves of zombies coming in )
[cheap attempt of comic relive]