It would take a very long time and it would not be pretty or pleasant but I believe it could be done.
Even in the event of an all-out nuclear exchange I would imagine a number of areas would probably survive relatively unscathed from the attack. Places like Tonga, New Zealand, Madagascar and Tasmania are all relatively remote and unlikely targets for nuclear attack. They are also in the southern hemisphere.
Most if not all belligerents in any nuclear war would be based in the northern hemisphere and missiles would be targeted at major cities, missile launch sites and other strategic locations mostly in the northern hemisphere. Although there would be plenty of contamination all around the globe the vast majority of fall out would be in the north.
If some of these smaller more remote countries were not targeted directly they would become centres for rebuilding civilization. Key features would be the survival of a working Government, some infrastructure and a knowledge base.
Even in the southern hemisphere nuclear contamination would become a major issue, but it would probably take some time to reach its maximal level by which point some of these countries might be able to put emergency measures in place to help protect people to some extent, impose rationing and put the countries onto a war footing.
After a hundred years there could easily be a flourishing technological civilization in the southern hemisphere even if it was still contaminated to some extent by fall out.