Something not everyone realise is that reactor designs can vary greatly and you can't really compare them safety-wise. If someone suggested to build a graphite moderated reactor like Chernobyl today I would be the first to protest, but you can't compare that to other reactor designs like the molten salt thorium reactors or pebble-bed reactors.
And despite e.g. Russia still having that kind of reactors in use, if you calculate the number of deaths because of nuclear, hydro-electric and coal per watt-hour produced, it becomes painfully clear that
nuclear is, by a fair margin, the safest power type we have. (You can use worst case figures for nuclear and still get the same results). If you factor in environmental effects like acidification and mercury poisoning from coal it becomes even more clear how much cleaner and safer nuclear is. Nuclear is more expensive than coal because coal doesn't have to pay for its negative side-effects (externalities) like nuclear do. The high cost of new reactors today is because the anti-nuclear lobbyists tactics is not to demand the reactors be shut down (primarily), but rather to lobby for stricter and stricter safety requirements, insurance-requirements and other tax based tricks to make them uneconomical.
Dealing with the waste is not a big (technical) problem either. It is very dangerous initially, for sure, but there is so little of it produced compared to how much power is generated that it is manageable and easy to contain. It is not like with coal which dump shit into the atmosphere continuously. Nuclear waste is solid burnt-out fuel you put into containers and store in a safe fashion, nothing gets into the environment. The "millions of years problem" is very misleading, it's true it will be radioactive for a long time, but radioactivity decline exponentially. In reality the hottest, most dangerous, components cool down very quickly; and after a relatively short time you are left with only the weakly radioactive stuff (with a very long half-life). So the waste will be slightly more radioactive than background radiation for a very long time, but not nearly as dangerous as claimed, and in fact, after a few hundred years comparable to radioactive ore that already exists in the ground in many places. Someone put it like this: even if 10% leaks out after 100 years you are no worse off than what you started with. So to simply deposit it deep underground really is a safe solution. (Even safer would be to deposit it into a subduction-zone in the ocean and let plate tectonics move the material back into the mantle, but that will never be politically possible of course. But just depositing it deep underground is still safe enough.)
Unfortunately we can't build enough new nuclear reactors to replace coal quickly enough to combat climate change even if there was political will to do so. I believe I heard that you can at most increase the capacity by 6% the next 30 years or so. Nuclear could be a small part of a solution though.
I don't think that the designs Ive seen offer any radical improvements. They are not immune to these problems as far as I know.
What about thorium molten salt reactors?