The TVTropes Trope Finder is where you can come to ask questions like "Do we have this one?" and "What's the trope about...?" Trying to rediscover a long lost show or other medium but need a little help? Head to Media Finder and try your luck there. Want to propose a new trope? You should be over at the Trope Launch Pad.
This is an inherent part of After the End, but the scope and timeframe of the events and story push this into various secondary tropes.
For instance, you have a Scavenger World when the apocalypse is relatively recent and/or the world is taking a while to recover, and the rebuilding is mostly factions squabbling over the remnants of civilization. (Mad Max and Fallout are good examples.)
You have a Cataclysm Backstory when the apocalypse was far enough back that it's considered a past event rather than something still happening. (Which usually entails at least some level of rebuilding outside of the bleakest settings)
A Points of Light Setting has scattered pockets of civilization separated by a hostile world.
From Cataclysm to Myth kicks in when the apocalypse is now in the distant past, with the exact details muddied over time ala Legend Fades to Myth. This can overlap with the others in some cases, with some groups having only a vague historical memory of the events and others having a clearer knowledge.
See the Apocalyptic Index for more elements that can mix and match here.

Is there a trope for when a world starts recovering after some sort of Dystopia, Apocalypse How, or After the End scenario? It tends to involve things like species returning from extinction or near-extinction, civilization being rebuilt, and the like; if there are no humans left at all, a different intelligent species may take over instead. Depending on how it's used, helping the world recover could be an Ending Trope, where it's likely to overlap with Surprisingly Happy Ending or "Ray of Hope" Ending. Alternatively, the world recovering after a recent apocalypse could be the main plot point from the beginning.
The only example I can really think of is WALL•E, where at the end, the humans who have been living in space for hundreds of years return to the abandoned Earth and help it rebuild (which is further shown in the Creative Closing Credits), but I feel like I've seen it a lot.