Nuclear power is just not that scaleable when you think through the implications. To stop climate change, you'd need a huge multiplication of nuclear industry. You'd need plants everywhere. All major cities in the industrialized world. And in developing countries. All of them, including those you wouldn't trust to officiate a child's birthday party without corruption and mismanagement.
All of that assumes none of those countries start using their new nuclear industry to get nuclear weapons. The world is unsafe as it is with some rogue/rogue-ish states having them. And dirty bombs are even easier, especially with all the nuclear waste being produced and stored... somewhere.
Either this or you need all the nuclear power stations to be in safe countries (and that definition might change suddenly) with an almost globe spanning grid. If you had a globe spanning super grid you could probably just do it with solar anyway, no batteries needed for the sun shines 24/7.
I don't think that either is good, but what you think is coming could potentially be delayed or stopped still. Once there is a nuclear meltdown the long-term damage is basically irreversible, not to mention the people that could die or have their lives changed which has a very real impact
All of that assumes none of those countries start using their new nuclear industry to get nuclear weapons. The world is unsafe as it is with some rogue/rogue-ish states having them. And dirty bombs are even easier, especially with all the nuclear waste being produced and stored... somewhere.