Getting enough food and water that wasn’t dreadfully contaminated alone would be the preoccupation of generations. By the time anyone had ideas about rising from barbarism, what we think of as civilization now would be rusted, eroded, and overgrown.
All of that assumes the most optimistic of assumptions regarding global wildfires, teratogenic effects, and nuclear winter. We wouldn’t be using magnetic tapes, we’d be using rocks and sticks and animal hides.
Remember we actually detonated a large number of nukes on the surface with minimal impact on global radiation exposure.
While relatively small by modern standards Yoshitaka Kawamot was less than 1km from the hiroshima blast and survived. Yes, these nukes may be 100+x as powerful but destruction is far from 1:1 with yield sizes.
15,000 warheads concentrated where humans live, on the fraction of the 27% of Earth that isn’t water. Be reasonable.
That said, a lot of that hardware, especially rugged, would easily last for a few decades.
We'd have to use cassettes for data storage once again.