Even if the populations of Australia, South America, etc survived mostly intact, the collapse of the global economy would push them back to the Middle Ages at best. Imports of food, oil, rare metals, tools, fertilizer... all those are gone and aren't coming back. If the nuclear fallout doesn't wipe out the population, starvation will push them back to Malthusian limits, and it'll be back to a hand-to-mouth agrarian existence.
It'd be interesting to see how much knowledge of the world before would survive the nuclear holocaust - eg, "I was an engineer before the wipeout, I understand Boolean logic, I know what computers can do, even if I have no means to build one myself". My guess is that knowledge would be spread too thin by our modern super-specialist economy and storage of it would be too fragmented for it to be able to bring society back, but at least the potential of people knowing "this was done before" could be there.
Also interesting, is that in a society where only 1% of people work in agriculture (that's a present-day figure, but I imagine it was similar 40 years ago), even basic farming knowledge and techniques are just lost on the average person - and like with the engineer example above, that knowledge may be spread too thin to be re-implemented after nuclear catastrophe. If we can't get productive farming going, the aftermath of WWIII would be less Middle Age and more Stone Age.