As far as I can find evidence of, there seems to have always been people in society who thought "the end is nigh", but not until the cold war/ "Duck and cover" era did they start to prepare for it. Sure the show "doomsday preppers" or whatever it is on nat geo is just a show, but i do knew a few people myself who have quite a few "bug out bags" and weapons stashed for the end they believe is coming.
From what I can tell, this mentality originated during the cold war. Do you believe this to be the case? Do you see those who obsess over bomb shelters as something new in your era or merely something evolved?
This is just a meta comment, but you really don't want experts of the Cold War here. You want experts on periods prior to the Cold War.
It is worth noting that pretty much all of the millenarians of the past — the people who believed the world was close to ending — believed it was going to be because of divine intervention, e.g. rapture or apocalypse or what have you. In such a situation a bug-out bag is probably not going to make the difference, or even be a good idea. What makes the existential threat of the Cold War unique is that it is entirely human-made, and thus needs to be "survived." The traditional religious end-of-the-worlders are not encouraging people to "survive" so much as repent, worship, what have you.
For pre-Cold War doomsday preparations, I would search out pre-Cold War doomsday fears that were entirely secular in origin.