That’s overly pessimistic, I think.
While we would indeed need to process seawater at an enormous scale, that becomes necessary after roughly 24 times longer into the future than the first nuclear power plant is into the past, which is when the non-seawater fuel runs out. This is too far into the future to make realistic predictions about technology or industry.
As for it being too late to stop global warming: well yes, but because that ship sailed some time ago. As is said of various things and time periods, the best time to do it was a few decades ago, the second best time is today.
The bigger problem is the political viability, both internally and internationally. How does Iran ever use nuclear power when every reactor is destroyed by an Israel afraid that Iran wants the atom bomb?
But, instead of thinking of it as all or nothing, think of the reactors than can be built as storage or intercontinental power grid capacity that doesn’t need to be built, or at least that we don’t need on the same time frame.
Right now, solar is doing what nuclear promised and failed to deliver for costs; batteries, while cheap enough to make electric cars interesting, are still more expensive for shifting day production to night load than just using nuclear to start with, and my preference of a global power grid needs a few decades to build even with (unrealistic) zero political concerns. All of this can probably be helped with nuclear.