There are multiple parallel universes, some of which have experienced nuclear war and some which have not
Maybe there are also many parallel Earth like planets in this particular universe, with human like creatures. Some of which have, some have not.
Could you elaborate on why? I'm not seeing why it couldn't be couched in similar terms (replacing the box with Earth, the radiation source with nuclear war, and the cat with people) -- is it because you can't treat the Earth as a closed system in the same way?
The problem is... why and how does your consciousness “move” to another reality? What is the physical mechanism that has you living and not a twin in another universe who isn’t really you... after all, you died when the bombs fell.
Most Of all though it’s important to remember that like Schrodinger’s cat this is a thought experiment meant To illustrate certain problems or features of quantum mechanics. It is an analogy or metaphor for a system dominated by quantum effects, like a single radioactive atom with a 50-50 chance of decaying in the next second. These thought experiments don’t really suggest alive/dead cats, or immortals, they highlight how quantum systems differ from Classical ones.
These ideas have led to various interpretations of QM, such as The Copenhagen Interpretarion (wavefunction collapse) and MWI (multiple universe). None of those interpretations are very rigorous, and may well be indicative of gaps in understanding as much as anything else.
Surprisingly enough, a human is much more likely to find oneself alive in a world in which humanity was not nearly made extinct by nuclear war.
Exponentially so, if we assume, reasonably, that even if humans survived (technically) a nuclear war, their descendants would take 100s of thousands of years or longer to recover to pre apocalypse levels, vs. non nuclear war realities where the population simply keeps increasing exponentially, leading exponentially many more humans to find themselves existing in that "timeline."