honestly I am too pessimistic about human nature
I think there would be a bit of a power vacuum for a while. The other thing to consider is that a lot of the US's nuclear arsenal would have been expended or destroyed. I imagine that there would be little political desire to see money being spent on expanding the arsenal, especially since their primary adversary has been destroyed and attention would desperately be needed on the domestic front. Long term you might even see a more pacifistic United States, the country until that point had never really suffered the ravages of major industrial warfare, now it had experience a war of nightmarish proportions. Ironically you might see a greater focus on institutions like government healthcare as there would be a need to treat millions for injuries and cancer.
There is likely a sentiment of "never again" and maybe a push to maintain a nuclear monopoly, but I feel that it would be quite a harrowing experience and a desire to never let such a thing happen again and a push for the reduction of nuclear arms across the world. To be honest its probably something we can barely imagine, such a catastrophic loss of life due to human hands has never been experienced and I cannot imagine the phycological trauma that would emerge for almost everyone involved. There is every chance that the opposite lesson could be learned, but I think the culture of the US would be fundamentally altered forever.
Its not a perfect analogy, but I think of gas weapons in WWI, following their extensive use, there was a desire to keep a limit on them to the point they were barely a factor in WWII, maybe you'd see a general revulsion at such weapons rooted in a deep fear of first hand experience. I'd probably imagine a reduction in the production on nuclear weapons globally, but not so optimistic as to think they'd go away forever.
Regarding other nations, it would be a miserable time. Europe would be a charnel house of nightmares. The population was around 600 million and most of them are going to die. It'd be lucky to have a quarter of that by the end. The USSR was around 200 million and that is probably going to drop to the low millions, it would be a patchwork of warlords, militias and localized communities. Banditry would be common and things would regress massively, likely to pre-industrial levels. You probably wont see any significant polities in this area for a long time.
China would probably be mildly struck by the USSR and be reeling from that. Japan and Korea probably takes a few hits due to its proximity to the USSR and alliance with the US. Korea would probably be heavily impoverished, Japan would be knocked down for a while too.
Part of the world likely untouched include India, Oceania, Africa and South America. You might actually see the US interfering less in these regions as they withdraw inward, so maybe likely a very different South America. I imagine in places like Korea and Japan, the US would be seen extremely negatively, regardless of who shot first. Europe might be slightly different if the US offers aid, but I don't imagine the Stars and Stripes would be welcome in many places. China might be well placed to become a dominant force in Asia, as it would likely managed to absorb what few hits the USSR would throw at them. Unsure how the Communist Party might survive, in such a scenario I imagine it's northern base probably takes a beating, with Beijing likely destroyed. Who knows how that might affect their grip on power in the long run.
India would be untouched, but the collapse of global trade is probably stunting. With Europe and Eurasia a burning wreck, there are bound to impacts of food that would hurt places like India and Pakistan hard. Despite this, I think India is destined for a slow climb into becoming a regional and eventually a global power with its population and industry intact. Who knows what political direction they might take, especially if the US is busy licking it's wounds.
Europe would be a patchwork of minor states. Any major state that managed to survive would be severely weakened and probably forced to adopt draconian measures to keep order. However I imagine their central governments would be unlikely to maintain control over their nations, even if some of its leadership was able to survive. Banditry and warlords would be the most likely scenario across the continent and I doubt any major polity would be rising from Europe for many decades. There would likely be a focus on immediate survival, and regionalist identities would probably spring up.
Places like Oceania would likely be spared, I imagine due to the relatively limited scope of the scope of the Soviet arsenal at this time, the focus would be on Europe and the USA. I imagine places like Australia and New Zealand would be trying to provide aid to the UK, but that would be a drop in the ocean in terms of what was likely needed. They're probably destined to become local hegemons over the Pacific.
Africa would be interesting. Decolonization was well underway by 1962, and I imagine what remained of European colonial structures would utterly collapse with their respective motherlands in turmoil. This hasty withdrawal or overthrow is likely to result in numerous ethnic conflicts however, not to mention disruption from climate events affecting crops, probably casing famine. The African continent is going to look very different, very quickly. South Africa is probably spared, but without the global condemnation of Apartheid, not being possible in this scenario, who knows that direction they will take.
