As long as you stay under 5%/minute or so, yes, you can with the latest generation of nuclear plants. Above that there are other alternatives that are more suitable (pumped storage, natural gas fired turbines).
And indeed, what you describe is exactly how it goes, nuclear power pretty much dictates the price of energy because it is the most expensive form. This is the big controversy of market based energy pricing to me. It should be the cheapest energy source that determines the price, not one of the most expensive ones.
I've recently looked into this in some detail and the situation is actually a lot more complex than you might think looking from the outside in. This is an interesting article on how energy markets arrive at their price (it is incomplete, and a bit dated, but still useful):
And indeed, what you describe is exactly how it goes, nuclear power pretty much dictates the price of energy because it is the most expensive form. This is the big controversy of market based energy pricing to me. It should be the cheapest energy source that determines the price, not one of the most expensive ones.