Load following with a nuclear plant isn't difficult if you can easily control the moderator. This can be controlled by computer. In designs with large negative temperature coefficients (such as LFTR) the reaction speed can be controlled by the rate heat is removed from the reactor, making load following is as simple as controlling the speed of a pump in a coolant loop. Most (all?) current commercial reactors are not designed to habitually operate this way.
Commercial reactors are usually run full power for capital cost recovery reasons. The cost of fuel for nuclear versus the capital cost of current reactors is such that it is always cheaper than the fuel (or storage) for alternative power generation, so in periods of low demand, nuclear wins. Capital costs are high because it is difficult to handle high pressure water (and the 1700 fold expansion in volume if containment is lost) in current commercial designs. Designs using molten salts operate at atmospheric pressure and will be dramatically cheaper to construct. Companies such as Terrestrial Energy and Flibe Energy are working on commercialization of molten salt reactors, which are feasible from megawatts to gigawatts. Such a reactor would be ideal for a remote research base.