You also can't talk about next-generation reactors because they don't exist right now (at least in production).
What I'm trying to say here is that this doesn't help our pro-nuclear case. It is cherry picking post hoc analysis. Climate change would still be a real problem, even with our leg up. Because frankly, climate change is much harder than getting solar, wind, and nuclear everywhere. It is also bigger than America, the EU, and China (total of 54%).
Are you referring to HVAC cooling (refrigerators, heat pumps, etc.), or reactor cooling (cooling tower, etc.)? Neither has any intrinsic carbon emissions though. HVAC needs electricity[0], but has no emissions other than incidental CFC release[1]. Nuclear / industrial equipment cooling typically needs water, but doesn't release any carbon.
[0]Ostensibly nuclear in the context of this thread.
[1]Mostly a non issue post Montreal-Protocol
We could use more electricity in all of those areas.
Next-generation nuclear reactors will operate at much higher temperatures and be more economical as a result. Some of the high cost of nuclear energy is because low-temperature operation is expensive. A PWR has huge steam generators inside the confinement vessel because using hot water to boil steam and spin a huge steam turbine is wasteful.
Next-generation reactors will have a gas turbine powerset that fits in the employee break room of the turbine house. Next-generation reactors will make hydrogen for fuels and fertilizer using processes like
https://en.wikipedia.org/wiki/Sulfur%E2%80%93iodine_cycle
There is no reason the rest of the world couldn't follow if the "Union of Concerned Scientists" wasn't afraid of a Muslim Planet. (It was Pakistan getting the bomb that put the kibosh on advanced reactors are sustainable nuclear fuel cycles.)