This might be the silliest argument I have ever heard that nuclear power is not a zero carbon energy source.
But have they considered the carbon emissions that would result from the burning of the world's cities if, say, the expanded use of nuclear power led to increased nuclear-weapons proliferation and, hence, to nuclear war?
But nuclear power takes by far the longest—anywhere from ten to 19 years from planning to operation—so it has the highest level of opportunity-cost emissions. When you combine these implied emissions with the emissions produced in the process of building the plant and mining the required uranium (plus the off-chance of a big carbon release resulting from a nuclear war), nuclear power produces as much as 180 grams of carbon-dioxide equivalent per kilowatt-hour. That's still a lot less than coal. But for a supposedly carbon-free power source, it's pretty lousy.
Every single country that currently has nuclear weapons, made their first nuclear weapon before they made their first reactor(North Korea might be an exception to this but they have so far failed to make a bomb that doesn't fizzle, despite having nuclear reactor experience). Therefore there is little reason to expect that blocking countries from building nuclear reactors will block them from getting weapons.
You cannot build a nuclear weapon out of reactor grade Uranium, or even Plutonium for that matter. Because of the requirement of separation of different isotopes before you build a weapon from either source, it is just as difficult to build a bomb out of reactor grade material as it is from natural Uranium. Therefore nuclear reactors do not lead to it being easier for terrorists to get their hands on bombs. The biggest risk with terrorists will always be a group of them stealing a weapon from a military not building their own(usually its Russia people are concerned about, but you could imagine a situation where a quick strike might be able to even seize an American one, although it would probably require hundreds of terrorists an event almost impossible to coordinate).
Now, there will be more highly trained people in nuclear technology if we build more reactors and this will make it slightly easier for nation states to build atomic weapons. However both Pakistan, and South Africa have managed to figure out how to build atomic bombs. These are not exactly the most rich, or technologically advanced countries. That demonstrates that almost any country seriously considering building nuclear reactors could have nuclear weapons if it wanted them, without going through the trouble of building the reactors as a stepping stone.
As for the second argument he makes, that the longer construction times increases the amount of Carbon produced based on opportunity cost. I don't really buy that either. First off, assuming a time of 10-18 years is silly, in any event where nuclear power will make a difference in America reactors will be built in six years or less. The actual construction time is in the 4-6 year range, the rest of that is regulatory red tape that will be easy to streamline without compromising safety.
Even assuming that his timing is correct however, it doesn't change the fact that this is a fictitious cost that will only be incurred at the start of a big scale up in the use of nuclear power. It doesn't change the fact that the total amount of CO2 produced by a nuclear plant over its lifetime is similar to the amount a wind plant of the same capacity would produce. Once the grid developed to the point where nuclear plants were replacing other nuclear plants rather than coal plants this cost would entirely go away.
Oh, and I get to the end of this absurd analysis to see his result. Even after he assumes the burning of major cities from nuclear wars, and adds in several fictitious costs that will not matter in the long run: Nuclear Power still produces electricity while producing one sixth as much CO2 as using coal plants.
Yes, the technology is that good, even those who try really really hard cannot make it look worse than coal.