There is not unused power from nuclear power plants. They are base loaded plants.
It is true that most nuclear power plants are base loaded precisely because if they were designed for peak load they would not be able to easily "throttle down" and would produce excess power. Because of this, utilities must have additional, less efficient power plants to add power to the nuclear plant when load is greater than base. If a nuclear plant was built in conjunction with a hydrogen production plant, the nuclear plant could be built to efficiently provide peak power without any additional generation from other plants necessary, and when off peak, which is most the time, the excess power could be used to produce hydrogen.