@James, ignoring for a moment the carbon cost of building nuclear power plants, and of mining uranium, and/or reprocessing soent fuel.
Nuclear is cheap because decommissioning costs are not factored in. 1000s of years of isolation for highly radioactive components makes decommissioning a nuclear power plant incredibly expensive. Never mond the fact the huge amount of government funding was invested to get the plants you have at the moment.
I'm willing to bet that on a fair basis, watt for watt, measured on a full life cycle, that solar power is currently cheaper than nuclear, or that with similar levels of government investment it would be withon 5 years (and China is currently investing).
Second nuclear power plants are great for base load. And not so great for varying loads. In the US max load is in the summer time powering loads of air conditioners (which i cidentally heat up the air outside your buildings, and make cities even worse hot spots... But that isn't a large climate issue, just a local one afaik). Nuclear can't be readily turned off (decommisioning plants take months and recommissioning them is expensive). So you need to look at base load - the minimum demand during the winter, and plan to have a maximum production of nuclear to supply only that demand.
Then you can have other systems (like hydro) which can be turne on and off at will, to supply higher demands (though there isn't enough hydro to make much impact, and all the low hanging fruit is already massive hydro projects).
Third, nuclear doesn't help address transport. We don't have decent/cheap alternatives to fossil fuels for carrying power around in a car. Nevermind all the rest of the transportation industry (UPS? Freight (lorries and trucks)? air freight? Train freight? Shipping?) the vast majority of the world depends on their food being shipped to them, and nuclear doesn't help solve that issue particularily well at all.
So in short, it isn't all about watts per cent. Free markets have never existed, and political there is some question as to whether they would work now.
History has shown some people perfectly willing to buy out the competition and then simply ignore it (new technologies which might threaten the position of oil and coal would be patentable, and pil giants would love to buy those patents and refuse to use them...) that is perfectly plausible in your 'free market' solution - and yes, is the effect of a government enforced monopoly (ie the patent system).
I don't see how regulation and/or public subsidy is any better or worse than enforcing a patent system. Unless the state is enthralled to large corporations who only have any interest in their own profits rather than the public good.