Yes, on a synchronous grid. Yes, everyone does it. Yes, it works.
You seem to be thinking in a binary manner, the plant is either on or off and there is some big central regulator doing the switching. But that isn't how it works.
Plants, through their grid connection, 'see' the grid frequency of 50.00Hz. If the frequency now deviated to 49.99Hz because consumption increased, the generators will be slowed down a little. Not a lot, because they intentionally do have some momentum reserve for very fast variations. Now that out generator has slowed, your nuclear power plant (say a BWR) reacts to this: a slower generator slows the coolant flow to the reactor. The load following will turn up the coolant pumps, cooling down the reactor a little more, leading to less steam bubbles, more moderation, more fission, higher power output. This higher power output leads to higher steam throughput, speeding up the generator.
If grid load decreases and frequency goes up to 50.01Hz, the reverse happens. Faster generator means higher coolant flow. Load following control slows down the coolant pumps a bit, steam bubbles increase through higher temperature, less moderation, lower fission, lower power and steam production, slows down the generator. Things are designed to stabilize at 50.00Hz. The exact amount of power the reactor produces is dynamically scaled up and down to keep the grid frequency stable.
For a PWR, there is some more control logic involved because steam bubbles aren't normally present in a PWR. So load following works by changing control rod positions depending on the grid frequency, which isn't quite as fast, but still in the order of minutes.
So load following is very gradual and non-binary. The myth that you can only on/off power plants is just a myth for almost any kind of plant.