A world where we produce N watt-hours of energy without nuclear plants and a world where we produce N+K watt-hours of energy with K watt-hours coming from nuclear has exactly the same effect on the climate.

Unfortunately no, this is not how it works.

The relative quantity of power provided by nuclear (or renewables, for that matter) is NOT our current problem. The problem is the absolute quantity of power that is provided by fossil fuels. If that number does not decrease, then it does not matter how much nuclear or renewables you bring online. And nuclear is not cheaper than fossil fuels (even if you remove all regulation, and even if you build them at scale), so it won't economically incentivize taking fossil fuel plants offline.

Nuclear cannot "make the climate better" but can perhaps slow the destruction down, only if it replaces fossil fuels, not if it is added on top due to increased energy consumption. In that case it's at best neutral.

Nuclear and everything we were using before is probably not better than just everything we were using before. Hopefully we can continue to reduce high emission or otherwise damaging power production even while power requirements grow.

Only if they build the reactors then go bankrupt leaving the rectors around for everyone else. Likewise if they build renewables to power the data centres.

Almost all energy consumed for computation turns into heat. Increasing energy consumption therefore doesn’t help the climate, in particular if you don’t source it from incoming heat from the sun (photovoltaics or wind energy).