The current selection is A Canticle for Leibowitz, by Walter M. Miller. It consists of three parts, each of which was originally published as a short story in 1955, 56, and 57. The three were then reworked and published as a novel in 1960. Canticle is a post-apocalypse story, set after a more-or-less undescribed nuclear holocaust. Miller wasn't the only one writing post-apocalypse novels (or short stories) at that time: George R. Stewart wrote Earth Abides (first winner of the International Fantasy Award; the apocalypse in that book was a disease that killed almost everyone in the world -- well, in the US,-- not a nuclear attack) in 1949; Nevil Shute's On the Beach came out in 1957; and Pat Frank's Alas, Babylon appeared in 1959. For a very extensive bibliography, see Nuclear Holocausts: Atomic War in Fiction, by Professor Paul Brian of the Washington State University at Puyallup.
Nuclear annihilation (by the Soviet Union -- back in the day, the US would not have started a war) was a major fear in the late 1940s and throughout the 1950s, hence the number of novels and short stories using it as a theme. Reading recently Mary Roberts Rinehart's mystery, The Swimming Pool, published in 1952, I was surprised to see a passing remark that the narrator was able to sell her family's summer home in upstate, rural New York because the buyers were worried about an atomic attack and wanted to get their family out of New York City. That was a throw-away line, very matter-of-fact. For a taste of the culture of the time, try Conelrad Central, a nifty site with too much material that I remember: 1950's "duck and cover" drills in school (in the event of a nuclear attack, curl up under your desk -- today the same drill is for earthquakes,) but also, Our Friend the Atom, a Disney Studios filmstrip and book. (I guess if it hadn't been for the Soviets, the atom would have been everyone's friend.)
Miller's atomic war, which occurred six hundred years before the novel opens, is referred to as "the flame deluge." Perhaps it's an allusion to Thomas Nashe's line "brightness falls from the air," with the accompanying refrain "I am sick, I must die / Lord have mercy on us." The Wikipedia article on Canticle (see link above) notes that Miller participated "in the Allied bombing of the monastery at Monte Cassino in World War II," and he wouldn't have been the only one to equate bombings with that line: Time magazine used it to title a June 8, 1942 article on the bombing of Germany.
Following the flame deluge, the "simplification" occurred: the destruction, by the survivors, of technology, books, and the people responsible (scientists) -- there's an implication that the backlash extended to all those who were educated. Somehow this seems less fanciful today than it might have in the 1950's: "let's kill the scientists, and all those intellectuals, and people who read for any reason except belief." Well, the future may be coming faster than we think, and, of course, not always in ways we expect.
Earth Abides had the same disappearance of knowledge, but it was gentler: the survivors focused on in the book simply didn't, or couldn't, learn what the main character could teach, and reverted over the decades to a happy ignorance. I read that book around 1960, and I have a memory of a scene where the main character and another person walk by the UC Berkeley Doe (the main) Library. The other person (maybe a child or teen) asks if there is information in there that could help them, and the main character says there is information, but it is beyond them all. Stewart, the author, was a professor at UC Berkeley, and, given that the main character was a grad student, I wonder how much that failure to pass on information reflects his views of the "next generation" of college/university professors.
A recent non-fiction book takes Earth Abides one more step -- it discusses what would happen if all humans suddenly disappeared, without any nuclear war. The World Without Us, by Alan Wesiman, is fascinating. He uses specific locations around to the world to discuss the probable effects the disappearance of humans would have, and has some very interesting examples of areas that have, in fact, lost all human population: Chernobyl, and a strip of high-rise hotels in Cyprus. On his homepage (click on the title link above) there is also a link to a USA Today article on the Lower 9th Ward in New Orleans, with a spectacular photo of a house now almost invisible for the foliage covering it -- that has to be kudzu. World Without is great background reading for any apocalypse novel: it's what would be happening in de-populated areas, regardless of the cause of depopulation -- war, disease, politics.