What movie had the first depiction of an anarchic, post-apocalyptic dystopia?

A good point!

Boy I thought of Metropolis (1927) immediately but maybe not post-apocalyptic dystopian enough?

I say it doesn’t qualify as “post-apocalyptic” – just future dystopia.

There is a contemporary variant, though it wasn’t adapted to film until after some of the examples already given. Karel Capek’s 1920 play R.U.R. described a robot takeover of Earth with subsequent elimination of humanity.