The great thing about nuclear power is that it's pretty safe and pretty clean, especially compared to fossil fuels and coal.
But the big problem with nuclear power is that it literally can't solve our short-term energy problems in time to make a meaningful difference to climate change, and it can't solve our long-term energy problems either because it still relies on a non-renewable fuel source that must be mined and that will someday inevitably run out.
In the short-term, nuclear power can't save us because nuclear power plants are extremely expensive and time-consuming to construct. Even in the most optimistic hypothetical scenario, it would currently take 4 to 5 years to construct a new nuclear plant. A more realistic non-hypothetical time based on real-world examples would be more like 7 to 8 years. And that's not including time required to overcome regulatory and political hurdles. We simply don't have the time to build all the plants we need, because we need them right now.
And in the long-term, nuclear power won't be sufficient indefinitely. It still requires fissile fuel (typically uranium), which is a non-renewable natural resource that must be mined. Even if we're willing to accept the environmental cost of mining, we can't get around the fact that Earth will eventually (in a matter of centuries) run out of usable fissile material.
If it were possible for nuclear power to delay or reverse the effects of climate change in the short term, then the long-term problems might be worth the short-term benefits. But given that nuclear power can't help us now and will still incur a debt against future generations much like fossil fuels do (albeit a less costly one), I don't think nuclear power is worth significant investment.
Solar, wind, and hydro all have issues of their own, but they actually can solve both our short-term and long-term problems. I think that's where our efforts should be focused right now. Nuclear power is a costly quagmire.