The Flintstones is debated whether the characters are truly prehistoric or post-apocalyptic, trying to mimick past modern conveniences.
It has been established that The Jetsons is set in the future of The Flintstones. Then again, this could simply mean that the population in Flintstones eventually rebuilt themselves to their former glory. The fact that the ground is a wasteland in The Jetsons seems to support the idea of an apocalypse.
An oddly common internet joke is that the people in the Flintstones are some sort of low-caste part of the Jetsons society...This joke originated on 4Chan, and is usually done with someone asking what planet the Jetson's buildings are standing on, which someone else replies with an angry image of Fred Flintstone yelling "Fuck those rich assholes and their magic sky castles!"
Unless the apocalypse in the Flintstones 'verse involved some sort of Jurassic Park-style genetic engineering that got out of hand, or the show is in fact on a different planet with species similar to prehistoric Earth species, the number of extinct species from very different geological eras makes it hard to argue the post-apocalyptic part.
To wit, the Flintstones celebrate Christmas. Take that as you will.
Surprisingly common in Saturday morning adventure series intended for children:
Parodied in Futurama, in which Fry, believing that he has somehow been frozen for another thousand years, finds himself in a post-apocalyptic world; as it turns out, it's just contemporary Los Angeles. In Fry's first millennium freeze, aliens in flying saucers came and leveled civilization on Earth (or, at the very least, New York (and excluding the cryonics building), from which it rose again. Twice! One of which was apparently time-traveling Bender's fault.
In season 6, Fry, Bender, and Professor Farnsworth time travel in a fast-forward-only time machine to the year 10,000 — After the End, in the sense that society has crumbled. They continue moving forward, hoping society will rebuild and one day someone will invent backwards time travel. We get to see society collapse several more times, for varying reasons including enslavement by giraffes, Robot War, a flood, an apparent ice age, and a parody of the Time Machine people. They keep going until they reach the year 1,000,000,000 — After the End of all life on Earth. They decide to keep going forward, as they really have nothing else to do, to see the end of the universe, billions of more years in the future, after the last proton "dies". Turns out time is actually cyclical, and the universe then restarts, allowing them to move "forward" to their starting point...which they proceed to miss, forcing the Professor to take them "around again."
An often overlooked movie called Rock & Rule (or Ring of Power as the chopped-up kids version is called) is about a world where, after a nuclear war, humans and animals merged together. The world has become surprisingly civilized, as most everything is done using rock music. The Big Bad of the film, a person known as "Mok" (a parody of Rolling Stones front man Mick Jagger) pilots a giant airship shaped like his face, and plans to throw a giant concert in which he hopes will raise the Anti-Christ.
Skyland was set after a point in which the entire world had broken up into sections, and required air vehicles to move between sections.
Adventure Time is set in a rather bright post-apocalyptic world, where an event called the Great Mushroom War wiped out humanity (and took a sizable chunk out of the planet itself) but brought magic back into the world, bringing all kind of crazy lifeforms with it. Finn is apparently the last human boy on Earth (or what's left of it)... apparently. Though earlier episodes merely implied this (and Word Of God confirmed it), there have since been a number of episodes about the event itself.
The civilization that produced the Xybers in Xyber 9 New Dawn ended in fire probably centuries before the series starts.
Debatable in Samurai Jack. In the opening sequence, it is revealed Aku, the main antagonist of the series, threw open a portal in time and sent Jack to the distant future, where Aku's evil "is law". Whether the setting is actually post-apocalyptic or not is debatable, but it definitely seems that was what the creators were going for.
Connections with other Cartoon Cartoon era Cartoon Network cartoons have been made though. The first episode has clues showing that Jack arrived in the ruins of The City of TownsvillePhoto proof. Considering the plot of Samurai Jack though, Aku's post-apocalyptic world would have been undone once Jack went back in time to stop him.
Samson and Sally takes place in a world where the environment is all but destroyed, and cities are sunk. And, for some reason, people have started whaling again.
Highlander: The Animated Series takes place after a meteorite hits Earth and sets off nuclear missiles (which were armed in preparation to shoot it down but the launch was delayed by an Immortal). Centuries have passed, and the world looks almost alien, with jungles covering much of it and strange animals present everywhere. After the collapse of civilization, the remaining Immortals have sworn to forgo the Game and keep certain scientific knowledge for when humans are able to make use of it again. Except for Kortan, who chooses to Take Over the World instead, ruling from the only city/fortress on the planet using remains of old tech to dominate primitive humans. The protagonist, Quentin, is the last of the McLeods (considering he's Immortal, there won't be any more) whose destiny is to find all the Immortals (now calling themselves Jettators) and collect their stored knowledge to help restore civilization. Naturally, Kortan is also searching for the knowledge.
Oh, and Quentin later gets to meet the Immortal who blames himself for the catastrophe. The Immortal is an expert astronomer who claimed that the meteorite would miss and that exploding nukes so close to the atmosphere would be worse, causing the military to trust him and not launch (but not disarm the missiles). Unfortunately, the meteorite diverts from the path plotted by the Immortal and smashes into the planet. Naturally, he's still feeling guilty about that.
One episode of Justice League sees several characters trapped in a parallel universe. It turns out that this world was destroyed by war, and the normal city the episode is set in (including this world's version of the Justice League) are being created by the mental powers of a mutated survivor.
Another episode features Superman accidentally being sent to a post-apocalyptic future where the human race is extinct. He meets the immortal Vandal Savage who claims responsibility and is so remorseful that he helps send Superman home so that he can stop his past self.
My Little Pony: Friendship Is Magic: The magical land of Equestria might be a post-apocalyptic world. This started as a fan theory (really more of a joke) to explain why ponies used objects that were not ergonomic to their anatomy (spoons, cups with handles, shovels, etc.). According to this theory humans nuked themselves and magical talking horses evolved in the radiation, then adopted aspects of human technology. However as the series progressed the after-the-end joke made more and more sense. For example the ponies use hydroelectric dams and enjoy modern appliances yet use carts, despite the fact that car tires for non-existent cars are buried in the ground.
This theory has pretty much been jossed with My Little Pony: Equestria Girls, as humans live in a parallel universe (albeit as an Amazing Technicolor Population with pony counterparts) entirely, though fans of the post-apocalypse theory are quick to point out that the Equestria Girls universe may not be our universe but merely an alternate one that has humans in it.
Cybertron of many Transformers continuities. The planet is often rendered barren and uninhabitable by the Great War. Nevertheless, there are often Cybertronians who desperately try to eke out a meager existence after most of the Autobots and Decepticons abandon the planet.
Visionaries is a cartoon set on a world where, a few years previous a mysterious force has causes technology to become inert, ushering in an "Age of Magic" where the society is medieval in appearance. Magic users - good and evil - become humanity's rulers. Some stories reference the previous Age of Science, many of the current characters having positions of authority back then.