There are types of nuclear plants that can load-follow much more easily, but it's not a design goal right now. The cost of nuclear energy is almost entirely construction and red tape; once you have a nuclear plant, the marginal cost of producing another megawatt-hour is so low that you want to run them as close to 100% as you can.
If more of our electricity came from nuclear, load-following would become more of a design priority. Or we would build some highly adjustable loads that can take a bunch of excess power when it's there, like aluminum smelting or electric car charging.
That's budgeted as part of the (still remarkably low) operating costs. In the US at least, anybody operating a nuclear plant has to pay a certain amount yearly into a fund for its eventual decommissioning. (The same sort of situation applies to insurance, in case anybody was wondering.)
It's weird and may relate to different Things being measured, but UK decommissioning seems to be vastly more expensive than US decommissioning. Judging by news coverage anyway.
There are many ways to store leftover capacity so that nuclear plants can operate all the time even when combined with renewable sources. For example, you could use leftover capacity to drive pumped storage plants. Pump storage plants operate at 70-87% efficiency [1] and can produce gigawatts of power on demand (in fact, within seconds). They are effectively the largest batteries in the world.
Other problems: you require a pretty significant hydrological infrastructure for PS: upper and catchment resevoirs, and a sufficient gradient to provide a net energy differential.
The logistics make multi-purpose use of PS systems (vs. typical flood-control or irrigation reservoirs) somewhat problematic as water levels may fluctuate dramatically over a brief period of time, and spillways/headways constitute a significant flash-flood drowning risk unless access-controlled.
Still, yes, pretty efficient energy storage systems overall.
There are already solutions for load balancing power. And these load balancing solutions for nuclear actually would be the same ones you'd need for wind and solar.