I thought the whole point of pumped storage was to use solar / wind to do the pumping and release the water back (thus running the turbines) when it was dark or not windy. Why are they using nuclear or fossil fuels?
Yes, that's another use. But when it is favorable economically speaking then fossil fuels and nuclear will be used as well. The pumps don't care what color the electrons are. It's mostly a matter of bookkeeping and less one of technology. And as long as demand for electricity during peak times does not coincide with the point in the day when the peak amount of electricity is generated this is a nice way to make some money for the operators.
The owners of coal and nuclear plants are the ones that paid for the pumped storage, they had the idea of making their plant operations more efficient, not of storing renewable energy.
This is one of the early large ones, built by the local utilities:
I am not sure about coal, but you can't easily shutdown a nuclear reactor, so there are times during the day when you are overproducing. So pumping some water is a way to not totally lose the energy produced during that time.
Shutting down a nuclear reactor can be done very fast (emergency shutdown, underload shutdown) but increasing the load takes much longer.
This is one of the reasons that the blackout of 2003 lasted as long as it did, due the rapid decrease in load a couple of nuclear plants did a rapid shut-down.
Most base-load generating plants are difficult to adjust to a rapidly varying load.
You could set a bypass around the generators to the cooling tower to reduce energy production without reducing the rate of nuclear reactions. They don't do this for a number of reasons which basically boil down to nuclear fuel being cheap so they would much rather shut down just about everything else.
This does create a minor issue if the grid fails, but that’s a rare event and doing an actual shutdown is considered safer.
Ah, so that's the reason. I actually wondered about this after the 2003 event, I was caught right in the middle of it and had plenty of reason for reflection on the theme of power generation (that plus a lifelong interest in renewables also caused me to look into the various alternatives).
"Neutron poisons" that accumulate in fuel rods during normal operation can also interfere with a re-start. Sometimes a shutdown core has to sit for a few weeks to let those reaction poisons decay before it even becomes possible to sustain a controlled chain reaction in that core again.
Nukes are actually much better at load following than is commonly believed. It's used a lot in countries without significant hydro production, like Germany and France, and load following capability is a regulatory requirement in many/most markets. For rapid anomalous load events in the grid you of course revert to the shutdown mechanisms, but the load-following is used to keep up with everyday demand variations.