The scenario in that novel can't actually happen. Even widespread radioactive contamination from a full-scale nuclear war at the height of the Cold War couldn't throw enough radioactive material into the atmosphere to cause acute radiation poisoning across the entire planet.

"The Silent Spring" has always been poetic license when it comes to nuclear weapons. We could cause a civilizational setback on an unimaginable, possibly unrecoverable scale for the human race...but we couldn't make ourselves extinct that way (for one thing: kind of a lot of people in Africa and South America, land of nobody's nuclear target). But the long term effects would be a slightly increased incidence of cancer and birth defects across the globe, but more people will die in the resultant famine from the collapse of global food production then will die due to direct or indirect effects of nuclear weapons.

Make no mistake: it would be the end of the comfortable 1st world lifestyle we know, possibly for good. But humans have existed on this planet for tens of thousands of years in circumstances far worse then now, and we have endured and continued reproducing regardless.

Playing around with https://nuclearsecrecy.com/nukemap/ a bit, it seems it would take a lot of very powerful nukes before any nuclear fallout would reach the south of Australia. Even with a fictional maximum kilotons (max for nukemap) nuke dropped in Darwin (north Australia) with maximum wind force south wouldn't make the fallout reach the south.