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.