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Caves aren't particularly friendly places to live with average 4-8°C and 100% relative humidity.
But this might be not even necessary in long term, read this https://www.quora.com/What-happened-to-the-radiation-that-was-supposed-to-last-thousands-of-years-in-Hiroshima-1945
Not that I'm encouraging nuclear conflict, but you have to be prepared for everything granted the amount of aggressive idiots at power...