Yeah nuclear costs the same, that’s the point. If you overbuild by two or three x to meet peak demand then per kWh you pay two to three times as much.
Doing that is part of the plan for the switch to renewables, and people are actively working on it. A friend of mine at Siemens energy for example is involved in such projects.
Let's overbuild a mix of renewables and nuclear, then use the excess power to do things like green hydrogen.
Or maybe we will find other uses when electricity becomes too cheap to meter.