Thank God. I dropped the whole P&S franchise once the story meandered over to nuking Argentina. Just saying, I think the Brits could have handled that better.*Puts on elitist monocle* Wait a minute, this is not Protect and Survive! Why herumph-hufflepuffle, this is most untoward.*Monocle falls into cocktail*
Nonwithstanding that of course a war in 1983 would have pretty much ruined the USA, I do think that just enough would survive that some sort of overarching political system would more or less reunite it, and that individual spots that happened by luck to be less damaged, combined with underground and/or greenhouse sealed type horticulture would provide for a scattering of oasis-like communities.
I would think life would be pretty harsh in these towns, and even worse in the wastelands (largely including what used to be major developed areas and the best land). But something that fancied itself still the USA would exist.
And although apparently the damage to Australia and NZ was low enough for them to recover pretty well, and South America is rather amazingly untouched, this reconstituted USA would be no minor power. It might not be able to live up to its prewar pretensions, but it would be a factor in both the Aussie and South American polities.
The way I figure the US would reconnect is, that although military facilities and people are prime targets in the war, at the same time only they are issued much in the way of equipment or training to survive. The vast majority of US citizens would die within months, but the most viable communities of survivors would tend to be those that manage some degree of organization, which would almost certainly be woven around some military basis--either scattered regular military people, or National Guard.
Although the stereotype of the crazy American officer or soldier who is some kind of religious fanatic and Bircher ready to shoot liberals dead on suspicion of Communism and/or smoking weed is pretty widespread and not entirely without some basis in fact, by and large American military people are more or less ordinary folks who found it in some degree worthwhile or necessary to sign up and serve. They are pragmatic folks. Postwar, the scattered survivors will have skills, equipment and materials not commonly available, and will probably try to bring about some organization and foster the recovery of as much normality as they can. This will generally fall far short of their desires of course. But when, by and by, some agent of a larger organization stops by, they are more likely than not going to be in uniform of sorts and hold a valid rank in the US military, and the scattered military elements helping organize, maintain and defend the little settlements will reintegrate back into the chain of command, in principle anyway. Although we might have some instances of crazy colonels or sergeants trying to make fanatical Gilead kingdoms or leading apocalyptic cults, on the whole the scattered soldiers will be trying to make life easier, and will be glad to be included in a larger network that may lead to useful trade and integration back into a bigger world.
Thus although the new organization of the USA is likely to be hierarchal and authoritarian, and will command resources that are both sparse and scattered, I do think the process of reintegration will go forward as fast as possible and the new USA will have some serious if modest projection of power. There would be all sorts of weird dissonances, with people living essentially in 19th century or worse conditions having occasional access to nuclear submarines, the occasional jet aircraft, and God knows what in some underground bases. The population base is low and the industrial capacity to maintain and renovate and possibly replace prewar high tech stuff is limited--but, I'm thinking, it will not be nonexistent. Due to random quirks in the Soviet attack pattern, and failure of all Soviet hardware to follow it, remarkably high levels of tech will survive here and there, and as the national network under military organization reforms, these places will get priority of protection and support, in return for being worked to the limits.
I would think President Bush would be resident there, having de facto become President-for-Life. For all my dislike of the Bushes politically, GHWBush surviving automatically makes him the universally recognized President (unless Reagan also survives a while, and Bush only succeeds to the office later) and the nucleus and catalyst of reorganization. Wherever the President is, is automatically the capital and headquarters. Surely he'll keep moving until a suitably developed location is found and upgraded, and the scattered surviving military hierarchy will crystallize around him. I don't like him but I do think he'd try to govern pragmatically and with an eye to the most rapid and thorough recovery possible.
So although the USA's resources are down to a tiny fraction of what they were prewar, those were of course tremendous, and the fragments left might be pathetic but only in comparison with the past. Compared to what is able to reach the region of the USA, it is still considerable, and being patched back together to recover strength. Since prewar Americans, in particular American soldiers, were scattered over the globe, Bush will be trying to bring them back home and put them to work rebuilding, and also reserve some interest in projecting American power overseas, with an eye to future generation's interests.
While some American refugees might be from pockets that simply weren't woven in to the national network before the people there had a chance to flee to Australia, others would be dissidents from the integrated culture, such as it is.
Now it is very unclear to me why the Australian Federation and the South American one should be at odds with each other. Neither one I suppose was in much of a position to poach on the other's territory or populations, so when they come into physical contact now, I suspect the exchange of trade goods would be quite mutually beneficial, so I don't see just what they'd be getting into a Cold War over. Neither has ideology that is deeply threatening to the other.
If anything, the threat that either a post-Soviet or post-Yankee organization might try to assert authority over the wider world will tend to unite them. Whereas I think Bush's policy to a non-Communist federation or two in the southern hemisphere would be to try to get good diplomatic and trade relations with both; there is no hope in trying to conquer either so the best thing for the USA is to trade instead. But the militarized recovery regime might seem creepy enough to both that they prefer their relations at arm's length.
I find it a little hard to believe the Soviets would not target anywhere in South America; if only to deny resources to the Yankees. Good thing they somehow neglected to do that!