IIRC the nuclear winter would not be as bad as once predicted, you could see a 10 celsius drop. That would still be enough to greatly reduce crop yields though.

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.