What was the first post-apocalyptic film? This is something I've wondered for a while. First, I should define what I mean by a "post-apocalyptic film". It's one which posits some great global catastrophe which shatters civilisation. It can show that catastrophe but the focus has to be on what happens afterwards: how do people survive, what problems do they face, can they rebuild civilisation in some form, or is it a struggle to hold on to what they've got? Nearly everything everybody took for granted has been swept away or changed out of all recognition -- social classes, political institutions, gender relations, fast food chains. People with guns have a big advantage -- until they start running out of bullets. And so on. Mad Max 2 and 3 are classic post-apocalyptic films (Mad Max itself is borderline, as it is interestingly set in a world sliding into chaos, but society is still holding together -- just). So is Threads, though it spends more time on the apocalypse itself. Children of Men arguably is; Dr Strangelove isn't, because it ends with the End.
In short, post-apocalyptic films show life among the ruins, and so should be distinguished from their near relations, apocalypse and disaster films, which don't attempt to show the long-term consequences of their particular catastrophes; though of course there is a grey area where the genres shade into each other.
I initially thought the first was H. G. Wells's Things to Come (1936), the middle section of which is unmistakably post-apocalyptic. Three decades after the start of a world war, fighting still continues, only now it's between the inhabitants of what's left of Everytown, and the tribes living in the hills, squabbling over a coal mine. An epidemic has killed half the population of the planet, but now that it is over, the town is recovering. Petrol is scarce, so a double-decker bus now serves as a butcher shop, and cars are drawn by horses, though people still wistfully remember how far they used to travel in them ...
But was there anything earlier? There's no reason why there couldn't be. Wells didn't invent the post-apocalyptic novel; that honour belongs to Mary Shelley. Her triple-decker The Last Man was published, anonymously, in 1826, and traces the fortunes of one Englishman as the rest of humanity succumbs to a plague. He ends up alone, wandering among empty museums and palaces, and then setting off in a boat down the east coast of Africa. As it happens, a no-budget version was filmed this year, though it appears to have traded the melancholy for large volumes of automatic weapons fire.
So, I turned to the venerable IMDb. This only has incomplete information for early films, particularly silent-era ones, but it's better than nothing; and it has a system of plot keywords, such as Post Apocalyptic and Last Man on Earth, which can be used to pick out likely candidates from before Things to Come. There are four in total, three American and one French. Actually, two of them, It's Great to Be Alive (1933) and El Último varon sobre la Tierra ('The last man on Earth'; 1933 -- though it's in Spanish it appears to be a US production) are remakes of The Last Man on Earth (1924). The catastrophe in these three films is a plague which kills only men; all men are wiped out, except one, who then has every woman in the picture competing over his affections. These three don't take the apocalypse very seriously, however: they are all comedies, and the later versions are musicals to boot. I doubt their makers were very interested in exploring what might happen to society should one sex die out (beyond suggesting that a female US president would allow the White House to be overrun by cats); they sound more like nudge-nudge wink-wink male fantasies of getting rid of all of the competition. (One link I found referred to the title of one of the films as It's Great to Be Alive When You're the Last Man on Earth, which says it all, really.)
The fourth candidate is Sur un air de Charleston (1927), a short film made by Jean Renoir. Here, the premise seems to be that a future war has wiped out Europe. An African airman lands in the ruins of Paris, sees a white woman, who proceeds to ... show him the Charleston. He learns to dance it as well. Then they fly away again. Oh, there's a chimp too. Well, I suppose it could be argued that it's some sort of commentary on the pervasiveness of American popular culture (not just the Charleston, but the African is played by an African-American dancer wearing blackface!) or an inversion of white anthropologists watching and recording indigenous dances, or something. But the indications are that it was just a bit of fluff which Renoir didn't even bother to edit into a proper film (that was done later). If there was a point, it was to show off his wife's dancing, and to play around with some film effects.
These all do appear to be post-apocalyptic films of a sort, but, at best -- and without having seen any of them, I must add -- they are amusing opportunities for seeing the world turned upside down, not serious excursions into the land of What If ...? In drawing such a distinction, am I just being a snob? Maybe it's just my own peculiar bias; for example in my own research I look for novels which treat the idea of city bombing seriously enough to have thought through the consequences of their suppositions. The authors think what they describe might really happen; so their readers might too. So I look for something similar in post-apocalyptic works too. But still, I'm happy to give the title of first post-apocalyptic film to The Last Man on Earth, for now; Things to Come can be the first serious post-apocalyptic film :)
PS To keep tabs on what's happening after the apocalypse, check out Quiet Earth.