We don't actually know that. It's a model projection. Academic models have a long history of being wrong and seemingly always in the direction of being too pessimistic, across a variety of fields.
Do we need to transition away from fossil fuels? Sure. Are the models so robust and so beyond question that nuclear should be ruled out on the basis of a handful of years of construction time? No way. The science is nowhere near solid enough for that.
Nobody serious is suggesting we do nothing but deploy nuclear reactors. Just in the energy sector, we should do a massive build-out of wind, solar, transmission, and, yes, nuclear.
> switching to wind+solar+storage and in the process democratizing energy production
These have large economies of scale too. While you might want to install a small propeller in your back yard, it's much more cost effective to get the energy from the grid supplied by a large scale wind farm.
> rethinking mobility (massive expansion of public transport, massive price reduction of public transport, massive investion in biking infrastructure, making outer city districts more attractive), putting a prize on CO₂ with a substantial steering effect (but ensuring that the proceeds of such a tax are given, in equal parts, to the population, so that people who contribute less-than-average to the climate crisis have more money available at the end of the day), transforming the system (because even with a prize for CO₂, there are lots of valuable things which cannot be measured in dollars, and competition pressure in unchecked capitalism deepens inequality and exploitation), ...
These may all be good ideas (and personally, I would certainly agree with some of those), but has nothing to do with whether the needed energy is produced by renewables, nuclear, or mass deployment of hamster wheels.
But at our current rate, the global CO₂ budget will be fully exhausted in about eight years.
Hence we need to seize other measures, measures which reduce our emissions on a shorter timescale: switching to wind+solar+storage and in the process democratizing energy production, rethinking mobility (massive expansion of public transport, massive price reduction of public transport, massive investion in biking infrastructure, making outer city districts more attractive), putting a prize on CO₂ with a substantial steering effect (but ensuring that the proceeds of such a tax are given, in equal parts, to the population, so that people who contribute less-than-average to the climate crisis have more money available at the end of the day), transforming the system (because even with a prize for CO₂, there are lots of valuable things which cannot be measured in dollars, and competition pressure in unchecked capitalism deepens inequality and exploitation), ...