I mean, during a nuclear war, the networks would only need to stay working until a counterattack could be coordinated and launched.
After that point, the devastation would be so large planetwide that I think any potential TCP/IP users roaming the irradiated wastelands looking for food while coughing up blood would have other things on their mind than network access in any case.
For smaller scale events, sure, but while the networking design might be robustness-focused, there is still a huge amount of auxiliary stuff required, these need power and cooling, maintenance and administration, repairs, and so on.
Some parts of the internet system might be robust and resilient, but the whole system certainly isn't.