Nuclear is certainly ready for climate change ahead. Nuclear is one of the tool to use to solve climate change. There is no doubt from the technical prospective. 1970 nuclear is still a tool, given they continue to extend the usable lifespan of the plan. This is not a technology problem: Palo Verde nuclear power plant works in the desert, by evaporating sewage water. This is not a problem at all for power plants near the sea.
I will be explicit: we should use nuclear to fight climate change, we should not destroy local environment to cheap out on building more carbon free power generation.
Nuclear produce tons on stable carbon free electricity. This stabilize electrical grid. Without it, we have to use something else to compensate renewable instability. Hydropower (if you are lucky), batteries (still expensive but becoming more and more viable) or fossil fuel plants.