So, a nuclear war happens and wiped out 80 % of humanity, along with all the cities. How would societies change after nuclear war?
There are no mutants.
There are about 150 raider gangs in North America.
The war used 42,000 bombs.
The Conditions for a Raider Gang to Exist Disappear as Soon as people Begin Organizing
Societies are not uniform, and how humans react to events and adapt culturally are not uniform or scientifically predictable either, but one can take a few generalities from observed history and extrapolate. With 80% of the population gone you can expect the level of technology to drop back to at least 18th or early 19th century levels by the 2nd or 3rd generation following the war with a few enclaves of more advanced areas. These enclaves will probably still not be quite as well off as before the disaster, and will probably not be focusing on developing new tech so much as preserving what they already have. (similar technological regression followed the collapse of Roman society)
You also should stop thinking raider gangs, and start thinking more along the lines of larger aggressive expansionist chiefdom's or city-states. Any group of aggressive people that manage to form anything permanent enough to last will usually quickly establish their own settled communities. It wouldn't really be mad-max style raiders anymore so much as warlike nation states similar to early ancient Greece with dozens of mutually aggressive city states competing for resources. Mad Max style psychos just don't have the sociability and technical skills to be anything other than isolated short lived nuisances to any community that manages to establish itself. Raider gangs only last for as long as it takes for a few hundred people to decide to form a militia and begin defending something. Its pretty much what did in the Viking raiders, image a small community of 600 or so people. For a modern equivalent to the vikings Imagine if zero law and order existed and a 100 man strong gang of hell's angel bikers descended on a sleepy 900 person rural town with maybe half a dozen cops and a few townsfolk with hunting rifles. The bikers could pretty much run the place and do/take whatever they want. Now imagine if the next time the same gang showed up it ran into a road blockade with 150 national guardsmen or similarly trained and armed town militia manning it.
Which brings me to my next point, every post apocalyptic story just kinda assumes nukes magically make the military go poof and disappear. There will still be at-least SOME military presence left, even if its just a few small national guard units, and their goal will probably be establishing well armed and organized communities to do exactly that. Put up a barricade with plenty of guys with guns behind it to stop these crazed outlaws from just coming in and taking whatever they want.
Realistically 'Raider Gang member" would not be a viable survival strategy after about 6 months. For maybe a few decades after that isolated outlaws similar to the ones in the wild west would make travel and trade difficult but really pose no real existential threat to any settlement bigger than a few dozen people. By which point said isolated and unique city state identities would be forming and maybe beginning to antagonize each other by expanding into each other's territory or access to resources.
Final Note: 42000 bombs would render the planet utterly uninhabitable. A 10,000 megaton exchange is about the limit we could have any chance of surviving, otherwise you are talking about destroying so much of the ecosystem that only insects, plants, and bacteria could survive. Even that would be kinda iffy since we start approaching nuclear winter issues at that level of atomic warfare. 1000 megaton exchange would be plenty to regress society without actually rendering our planet nearly or fully uninhabitable. After a 10,000 megaton exchange people would be too busy trying to find food to even form a society.
What happens after a war is not easily predicted ahead of time. The general pattern is that something small takes hold and grows, but which small thing takes hold depends greatly on the fine details of the war.
If the 80% are evenly distributed, then that means the US still has 60 million people. That's a lot of people, so we can probably ignore the raider gangs unless they're gigantic (like each one is multiple-thousands of people). If they are very large gangs, that might be something interesting to look into. Why are there gangs like that? What small pattern took hold and caused them to form these gangs, however big they actually are? How did they get enough weaponry/power to cow the remaining people into not organizing against them? Why didn't the majority of people arm themselves? Those are interesting questions that could form the framework for a book.
What's the one thing that would change for sure? I'd say they'd begin practicing fighting with sticks and stones. After all:
I know not with what weapons World War III will be fought, but World War IV will be fought with sticks and stones. -- Albert Einstein