So I should first explain why I think this is a very hard question to answer.
A complete nuclear war in the northern hemisphere would cause a lot of short term issues, but ultimately the southern hemisphere would benefit from this in the long term. Here's why.
Australia holds around 30% of the world's Uranium reserves. It's essentially the Saudi Arabia of nuclear fuel. Ironically, it doesn't have a nuclear power program and probably never will due to public sentiment. Even more ironically, it could really use one right now, but I digress.
The point being that if power ever became a problem in this new world, the power sources needed in terms of nuclear fuel could easily be supplied from the Southern Hemisphere.
The same is true for Iron (large holdings in Australia), Copper (Peru produces the vast majority of the world's copper already), and even Coltan (most of it is in the Congo so smartphone building is still possible). There are massive natural gas reserves in the East Timor Sea, and oil in many places through the Southern Hemisphere. Most importantly, the vast majority of consumers of these resources are now all dead, so they will support a smaller population for a much longer time.
Southern Africa, Australia AND South America are all natural food baskets, supporting agriculture to varying degrees. (Southern Africa most so of course.) Africa, Australia and New Zealand all have massive food export industries; New Zealand produces most of the powdered milk consumed on South Pacific islands and would continue to do so; Australia has a massive live export market for sheep and cattle but this would die off as many of the countries that buy live export domestic animals are now gone. There are very few foods that cannot be produced in the Southern Hemisphere, and certainly none that could be considered essential to life.
Technology and Learning
Education is actually Australia's 3rd largest export industry today (after minerals and agriculture). Australia has been directly responsible for many medical advances over the last 50 years and continues to punch well above its weight in other technical fields despite the challenges successive governments have had in maintaining a world class education system in a modern economy. It should also be noted that other Southern Hemisphere countries are starting to build their education capabilities and will continue to do so over the coming decades.
Lack of Ideological Conflict
There are some exceptions of course, but generally there are no countries in the Southern Hemisphere that really all out hate each other. There are certainly people who do, but that's easier to manage with no state sponsorship and a world destroying event changing public opinion to a more collegiate state. Generally speaking, Southern Hemisphere countries are going to find it much easier to get along in the long term and engage in healthy trade and information sharing for mutual benefit.
So what does all this mean? Well in the long term, you have technical knowledge, resources and state structures that would not only survive, but thrive if their countries were the only ones left in the world. Less competition for resources, more average self sufficiency among countries, and no critical shortages of anything that would make state actors upset and want to fight for an equitable share. Add to that no shortage of education and knowledge and you've got an enlightened, harmonious community.
But, they won't progress anytime soon.
They may not be as screwed as the Northern Hemisphere and their society may largely survive intact. That on its own doesn't mean that they will continue to progress in their technological capabilities or social development over a 500 year period. Let's look at some of the reasons why.
Short Term Impacts
No Internet, at least not for some time. All the DNS server infrastructure that manages the decentralised network traffic around the world is largely based in the Northern Hemisphere. A lot of copies of knowledge are going to be destroyed in this war, and not ALL of it will have a surviving copy on a Southern server somewhere. Even if it does, there's a lot of work involved in getting a Southern Hemisphere 'Internet' up and running, and it simply wouldn't be the priority of governments in the event of such chaos as one would expect from an event like this.
Bear in mind here that the internet is largely an American invention and much of the original infrastructure investment was exclusively from a country which no longer exists in this scenario.
Global markets have now literally disappeared in a bright flash, meaning that in the short term, the global economy and how different countries relate to it plunges into chaos until the situation described above becomes more apparent. Remember that many industrial ventures in the Southern Hemisphere are funded by Northern Hemisphere global companies, so foreign investment can't inject new capital into these Southern countries, adding to the economic chaos over the ensuing years.
Finally, there is the simple morale problem; many people in this day and age have friends and family all over the planet, many across the equator. This will be a massive drain on morale, and extremely fear inducing as well. Governments will need to put a massive effort into 'business as usual' messages, while still sorting out the fact that some people employed solely to cater for export markets are instantly out of work.
Ironically, while Southern Hemisphere countries will have massive access to raw materials, they won't have the manufacturing infrastructure to capitalise on it immediately. Australia (for example) used to have a thriving clothing industry that has effectively been shut down as we import all our clothing from China and Bangladesh. Many of those displaced workers supporting export markets will need to be retrained to domestic manufacturing tasks, but this again takes a massive investment in infrastructure that will probably (in the short term) need to be met by the State as private investors won't have capital anymore thanks to a global market.
Also, there will be a higher need for extreme engineering manufacturing; things like container ships to move around materials between the 3 major remaining continents and all the islands between them. Much of this has now been lost meaning transport ships get smaller again and countries revert to self sufficiency thinking as much as possible to cover off the risk of ships not being able to get trade goods around with the efficiency and magnitude of the past. Additionally, the risk to ships has just increased as they are more likely to want to stay in Southern Waters, making the Roaring 40s a legitimate trade route again. The losers in this problem are the smaller islands who would rely heavily on trade ships for supplies, but also on income (tourism).
Some industries (like GPS and communication satellites) simply cannot be supported or warranted with the number of people left in the world. Only around 12% of the world's population live in the Southern Hemisphere to begin with, so the sheer drop in scale of economies that relied on trade with Northern countries will impact some industries insofar as they are no longer commercially (or technologically in the short term) viable. The 12% of population remaining are also spread across 3 major continents and hundreds of islands, further limiting scale of industry. Certainly in the short to medium term, focus will be agriculture, reinstatement of domestic manufacturing industries, and THEN trade, research and technological advancement.
The aforementioned container ships are another example of an increasing scale that would simply become nonviable in this world. Also skyscraper architecture, biotechnology and maybe even epidemiology; our advances into medical science will take a back seat to keeping the existing population healthy on a much smaller budget.
The good news for the Southern Hemisphere is that after a generation, maybe two, societies in this world would stabilise. There would be far less war, less concern about climate change, and a network of countries that work together in a more collegiate manner than they do now, albeit because the survivors are now ALSO more self-sufficient than they are now.
The bad news is that they will find it MUCH harder to progress in terms of technology or even social structure because the focus over that time will be survival, not growth.
It's been a long time (several generations at least) since we've had to worry about something killing us when travelling, trading, or even just earning a living. This event would shift the focus back to that, while at the same time reducing the size of the global population by an order of magnitude. Both of these things are killers for growth and development, but not necessarily killers of society as a whole. So; if you want to destroy society, it might not happen. Limit development? Most definitely.