Firechat was used a lot during the Hong Kong protests in 2014. If you search for "mesh network app" you'll find similar apps as well.
Assuming a global outage there's nothing to be done though, as there would be no electricity for miners.
Assuming there's still localized pockets of miners running, you run into the challenge that the mesh network still needs to somehow connect to the rest of the internet, where the miners are reachable. This could work if there's still an internet gateway in your city. Near impossible if you're in a rural area with low population and thus mesh node density. As connectivity between miners degrade, chain splits become an issue as well.
So in a post-apocalyptic world without the internet, electricity, and maybe water, your biggest worry would be how to send bitcoin? Bitcoin can already work over satellite or via radio waves. But again that requires electricity or a power generator of some sorts.
Maybe water? Definitely water. Unless you start collecting rain, have access to a well or live near a large body of freshwater you won't have water either, as there would be no power for the water pumps.
