I know that most real-life nuclear warfare simulations leave Australia, Oceania, most of Africa and South America unharmed by actual bombing, since they don't have their own nuclear armament, and are not considered major threats.
I know that Fallout shows nuclear weaponry in a much more positive attitude, but I'd say alot of these countries/continents would still persist in not having a nuclear program. At the very least, I'd say Oceania, especially the Pacific Islands, would survive. At most, I'd say Australia would survive (It's already nearing its theorized maximum self-sufficient population, which is too small to support a nuclear program), as well as Africa, the southern half of South America, and areas of Africa, as well as Oceania.
NOTE: I'm counting New Zealand as part of the Australian continent.
Considering Australia is where a heck of a lot of uranium for said nukes comes from, and was the site of the UK testing sites, in a falloutesque world where START and SALT didnt happen, I can see Australia getting nukes... Back then they wanted them, but the UK and the US rightly didnt trust them with nuclear secrets (Australia was known to leak live a sieve at the time - I've heard that the Russians wouldnt bother getting Uk cabinet papers in London, it was easier to get them out of Canberra (ASIO, the counterinteligence department, is still a bit of a local joke there).... Sure its harder, but a homegrown nuclear program is definately possible, Australia has everything it needs to do it.
If instead the US and the UK looked to a big splotch of desert near the new threat (China) to hold a few forward bases (like the US had in Europe to curtail the USSR), those simulations would also look different.