OK - Let us address each concern in turn:
Battery reliability: Solar cells come immediately to mind. It would not take many, and you could set up for the current you need. Install with a rechargeable, and you are all set.
EMP: Those little CD Geiger counters are both incredibly simple and designed for survival. The mil-spec. ones are even better that way. A larger concern would be whether the counter could give you meaningful readings for Alpha and Beta particles - mostly harmful only if the producing materials were ingested or inhaled. Gamma radiation can be detected in any number of simple ways, no batteries needed.
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Note that Plutonium is primarily an Alpha emitter - and the basis for most nuclear weapons these days.
Dosimeters: Those dosimeters will be useful only if you *know* there is a danger, and therefore pay attention to them, not so much that they indicate (they will), but the period of time it takes them to indicate. Note that they are useless for Alpha and Beta particles, as they are too well shielded. A piece of heavy aluminum foil is sufficient shielding for either - but for the inhalation issue.
50 miles of an 'target': That would apply to most of us. And if one is within several hundred miles, but east of a detonation, the radioactive products will catch up to you in any case, unless you choose to not breathe.
Survival: No modern medication, including statins, blood pressure medicine, epi-pens, antibiotics, vaccinations, nor any other infrastructure-dependent materials after the immediately existing supply is used up. Note that these materials have a relatively short shelf-life, so stockpiling is impractical. Then it gets down to supermarkets, transportation, human predators and the other wonders associated with a collapsed civilization. If your survival is based on a FBO, you will be subject to siege - and that is 100% a losing battle, 100% of the time. History has proven that over and over. And if you are nomads, fuel (vehicles) or food (animals) become a serious issue. As well as being highly visible and high-value targets.
Best to work towards preventing Armageddon, rather than preparing for it. Spend that energy on something more joyful than the collapse of civilization.
Nuclear Plant Accidents: See electroscope above. If there is enough ionizing radiation released to do you harm, that little device will indicate pretty much instantly. Meaning that you will have to 'get out' along with everyone else, and have somewhere to go. At which point, only a Chernobyl-level event is any real threat outside of that 'magic' 10-mile perimeter, or 20 miles +/- downwind.
NOTE: We keep a mil.-spec. Geiger counter, as in the past, my wife was fond of Fiesta Ware, some colors of which are radioactive. Not so as to be harmful, but one does not wish to eat from them. And before the hospital I work beside closed down in 2019, the radiation-safety people wore dosimeters routinely as they worked around radioactive materials used for cancer treatments.
Peter Wieck Melrose Park, PA