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Humans co-operate, yes, but we also compete. It seems to me that if co-operation were such a strong human instinct as you hope, we wouldn't be having a discussion about nuclear war in the first place.
The funny thing about modern society is that in a way we are all working together, we all contribute to the machine that operates days after day in bringing us food, electricity, and entertainment. But in so doing we have isolated ourselves. The denser our populations become, the more antisocial the individuals within become. We live in flats with neighbours above, below, and on either side, but don't know any of them by name.
The presumption that the breakdown of society would result in violence and anarchy is presumably based on the assumption that it is only the provision of food, water, electricity etc that modern society gives, that actually holds the entire thing together. People who would have been in opposing tribes if left to the "natural order" live side by side, but it only works as long as the incentive is there. And in general terms, this holds true- In cases of power cuts and such we often see people reverting to this tribal sort of mindset.
In addition one has to consider the instinct for self preservation, in the face of almost undeniably inadequate resources. We can harmoniously share our resources only as long as there are enough resources to be shared. In an event where clean water is scarce and food even moreso, as it would be in any urban centre after a nuclear attack, people are forced to compete in order to ensure their own survival, and we're back to the start of the whole circle again.
Humanity is a young species and we are still far closer to our most feral instincts than we would like to admit. When push comes to shove, are you going to be the one who puts your hand up and says "Okay, I'll be the one who starves to death. Take one for the team, you know" or are you going to crack someone's skull open to take their share? The answer isn't the one you probably want it to be.