Firstly, without technical expertise and a loosening of safety regulations (which could potentially compromise the whole exceptionally safe thing) nuclear power plants are prohibitively expensive to build and maintain in many countries. Simply put, most western countries haven't built many nuclear reactors in the past few decades, and that means the infrastructure to do so efficiently, cheaply and quickly no longer exists. The problem with the "just switch to nuclear" argument is that energy policy kind of has to be decided a long time in advance. For the US in particular, that might not be a huge problem as the US has been exceptionally shit among developed nations at planning for phasing out fossil fuels. But then, the countries with a lot of expertise in building renewables are going to be a lot more willing to do business with the US than the countries with a lot of expertise in building nuclear plants.
And then there's the more general international and political problems with nuclear energy. The power plants themselves may not be dangerous to nearby civilians, but they are dangerous because they can produce substances which can be used as weapons. Not all reactors have that problem, but the technical knowledge required to build and maintain those reactors can still very easily be used for other purposes. Every single country which has developed nuclear weapons in spite of the non-proliferation treaty did so by piggybacking off a civilian nuclear energy program. There is a good argument for deliberately limiting the development of infrastructure and technical expertise related to nuclear energy.
Finally, many forms of nuclear fuel and waste are security risks, and need to be guarded and protected at least until they've decayed enough to not be dangerous and/or useful any more. Some types of fuel are also vulnerable to criticality or spontaneous ignition if not stored correctly, which again could potentially be dangerous. Disposing of nuclear material safely generally requires a long-term commitment to ensuring it doesn't end up in the environment or get stolen.
Solving the climate crisis, if that's even possible at this point before civilization as we know it collapses (I doubt it) is going to require an unprecedented level of international cooperation, and you could easily argue that all these problems could be overcome. I personally think nuclear energy would play a role in any future solution, but not in the sense of everyone deciding to build hundreds of nuclear plants. I think nuclear power will really help those countries which already have the requisite skill base and infrastructure, but it's not a universal solution. As the technology improves, it may be that new reactor designs will become small and safe enough that any country can build them at reasonable cost, and the fuel problem could potentially be solved by new recycling technologies, but right now it's not there.
