The core of a conventional nuclear reactor continues to generate a lot of heat even after it is shut down by inserting the control rods. It takes many weeks for the heat generation to decrease to the point where the cooling system can be shut off. The plant has to have backup power to keep the cooling system going, or the core will heat up and eventually melt. How fast it heats up depends on how much cooling capacity remains and how long the reactor has been shut down. The first few hours are the most critical.
After the tsunami, the problems at the Japanese reactors began when the flood water wrecked the backup power systems and they lost the ability to sufficiently cool the (shut down) reactors.
Then the plymouth reactors don’t have backup diesel/ natural gas generators on site? They get auxiliary power only from off site? Sounds odd.
In the US, containment buildings are designed to withstand a complete meltdown, i.e., to contain the hot gas. I was shocked that this wasn't the case in Japan.