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.