Anthropic selection is not magic. More precisely, it is the least amount of magic necessary to ensure our existence.

If a miracle is necessary for you to survive, but either a smaller miracle or a (much) greater miracle can do the job, then if you survive, you should expect that it was because of the smaller miracle.

Multiple independent improbabilities are multiplied, so similarly if you can be saved by either one miracle alone or ten independent miracles of comparable size, then if you survive, you should expect to be saved by the one miracle.

(It all adds up to the nearest possible approximation of normality.)

So my guess is that the argument implicitly made by Julia and/or Toby (I haven't read the book nor listened to the episode) could be that although it is technically possible that we have avoided nuclear extinction because of a sequence of lucky outcomes, when you multiply the probabilities, the result is so small that there is probably something else (perhaps also some kind of luck, but smaller in magnitude than this entire sequence combined) responsible for the outcome.

In other words, you should not accept the answer "you were saved by a miracle of a probability P" before you have sufficiently explored possible miracles with probabilities greater than P. (Even if you believe that you were saved by anthropic magic.)

When I try to think anthropically about nuclear extinction specifically, it seems to me that there are also possible outcomes other than "no nuclear war" and "humanity extinct". Like situations where (most of) USA and USSR (and a few other countries) were nuked, but many places were not nuked, people survived there, and even the nuclear winter or whatever didn't literally kill all humans. If there were many nuclear "near misses", then there should be also many alternative histories like these. After enough such branches, even with anthropic reasoning, the probability of finding ourselves as survivors in a post-apocalyptic world is greater that the probability of finding ourselves in a world without a nuclear war. Not sure about exact numbers, though. The opposing argument is that the world without a nuclear war contains much more humans than any individual post-apocalyptic world.

Nuclear war doesn't need to produce human extinction to cause anthropic effects. In the world with such war there will be (presumably) less universities and less people who are interested in anthropic, as most such centres will be destroyed during nuclear exchange, and people will be more busy in survival. 

Also global internet is less likely to exist in the world after a large scale nuclear war, which means lesser exchange of the ideas and lesser chances for the things like LessWrong to exist – or smaller number of people participating in them.

I guestimate that in the world after 1-billion-loss-nuclear exchange where will be 10 time less people interested in anthropics.

Thus it is not very surprising to find oneself in the world where large scale nuclear war never happened, but it is not evidence that nuclear war causes extinction.

I agree with you that observational selection effects are important here.

Our causal model of reality tells us that if there were a nuclear exchange between the superpowers, there is a good chance we or our ancestors would've died, and once dead, would've been unable to engage in this conversation (or to produce offspring to engage in this conversation). This information needs to be taken into account in certain situations, e.g., when estimating the likelihood of a nuclear war in the future.

In particular, if nuclear weapons suddenly appeared for the first time today, we would have some guess, some probability of their being used in anger (in the future). Our knowledge that they've existed since 1946 without having been used lowers that probability. I.e., it is evidence against their being used in the future, but it is weaker evidence than it would have been if we were immune to the effects of the weapons, i.e., if we were equally able to observe their past use as we are their past non-use.

Moreover, suppose you, your parents and your grandparents were dependent for their income on their ownership of a sporting-goods store in Washington, D.C. D.C. is of course a likely to be targeted with many bombs in any nuclear war between the superpowers, and any owner of a sporting-goods store is likely to stay in or near the store whenever looting is likely (at least if their family's income depends on the store). Then the fact that there was no nuclear war between 1946 and today is less informative to you than it is to someone whose ancestors lived somewhere unlikely to be targeted (e.g., New Zealand). The person in New Zealand can of course communicate their observations to you, but if you're being sufficiently rational, you have to consider their communications biased in the same way that you have to consider the communications of, e.g., a drug company to be biased if you suspect that the drug company is likely to withhold information that shows their drug to be harmful.

I think it's only worth considering anthropic selection as a meaningful hypothesis for survival probabilities that are less than about 10^-9, and where we are very confident about that upper bound. Nuclear war doesn't come anywhere near meeting either of those criteria.

I do think we're in one of the few worlds that made it thru the Cold War intact. I consider our existence weak evidence for Many Worlds and quantum immortality.

Gives a reason to be subjectively optimistic about the future. But, I hope, not complacent. I find it hard to shake off the notion that the number of worlds in which civilization survives matters. But I also find it hard to explain why.