Climate disaster? To who? Humans? Mother Nature, on average isn't concerned. She adjusts. She evolves. History is clear on this. Radiation on the other hand is all but permanent. Climate disaster is a self-inflicted death blow by humans to humans.
No it isn't. The purpose of a nuclear reactor is to take highly radioactive material found in nature, cook off a bunch of its radioactive energy (producing power in the process), and create less-radioactive material as a by-product. The waste that comes out at the end is less radioactive than the material we started with, albeit much more concentrated. But that makes it easier to handle and store, compared to the CO2 and other greenhouse gases that are spewed out all over the place.
If we dumped all that radioactive waste into the ocean (BAD IDEA), mother nature would dilute it for us and spread it out until it disappears into the natural background radiation. The drawback is that it'll be dangerous to everything that encounters it before it's diluted, so we can't do that.
Also there’s the pesky fact of half-life. Ignorant anti-nuclear activists seem to think that nuclear waste is extremely dangerous and lasts for centuries. Nope. Nuclear material can either be imminently dangerous or have a centuries long half-life. It can’t be both.
This is not true. Nuclear fuel not highly radioactive. Their use in nuclear reactors stems from the fact that they emit two neutrons when hit with one, producing a chain reaction. Their byproducts are much more radioactive than the original fuel.
As a human that's the particular sort of disaster I care about most.