Last time the world had that many people in it was around 1750. How fast could the people back in 1750 get to the present technological level, if they had all the science and engineering and exploration already figured out, and also got a bunch of machinery to kickstart things?
That might still leave around 4-500,000 scientists worldwide and millions of engineers, millions of medical doctors, tens of millions of technicians of different kinds. Sure, many of them will be in odd places like New Zealand, South Africa or Argentina. But that's likely where most survivors would be as well.
Just for comparison, until the 1940s the US and Europe produced around 10-20,000 doctorates per year. At no point back then did we have more scientists worldwide than we would have after 95 percent of scientists were killed today. And unlike back then, the surviving scientists would have knowledge of every scientific knowledge produced afterwards, either as personal memory or in books and databases.
It's quite possible the internet would survive as well. After all, it's designed for that kind of events. Thousands of libraries would survive. Millions of machines, vehicles, airplanes, computers, components etc. would survive. Wikipedia, the 100 most important scientific journals, the US patent organization database can easily be on a laptop and libraries around the world are making copies all the time.
Similarly, some refrigeration is not super difficult to build once you know how it works. That would help quite a bit against starvation from bad harvests because it allows you to have bigger stores of food.
With everybody struggling to raise crops and livestock, let along trying to reinvent the light bulb, it would a long slow climb back to civilization.
People aren’t going to forget how to trade. The people with that manufacturing and engineering knowledge aren’t going to be busy with livestock. They’ll be trading these valuable goods for food.
I suppose refugees and the political vacuum created would be significant problems.
If you couple reduced crop yields with millions of hungry refugees, economic instability and general chaos... Who knows. Unlikely to result in human extinction, but very painful for everyone no doubt. I do think it seems unlikely that people would nuke Australia and South America. I honestly hope that if nukes are ever fired, as many countries as possible remain untouched.
Have you ever heard of https://en.wikipedia.org/wiki/Pine_Gap ?
While that is only one target, and in the middle of nowhere at that, do you still think so? For me that looks like a very valuable target to disrupt the operations of anyone who operates from there.