Here is an article that explains the initial purpose of the pre-internet, that is, the ARPANET. It really does not address portions of the internet surviving Armageddon.
See: https://www.wired.co.uk/article/h-bomb-and-the-internet
I worked in a store and forward message switching network, the AUTODIN, for many years. See: https://en.wikipedia.org/wiki/Automatic_Digital_Network
and images for some of the terminal and switching equipment configurations --
https://www.google.com/search?q=AUTODIN&tbm=isch&source=iu&ictx=1&vet=1&fir=WiJQI4s7p1onYM%2Cxz23dO8bL9WxhM%2C_%3BouaYkMF-l3ptcM%2CZd97KOpnYDeVpM%2C_%3B93KATQLz-qOObM%2C_BQJyHuUWIrXwM%2C_%3BbKtITbN_H5WJDM%2CZd97KOpnYDeVpM%2C_%3BwwWh5QBngHrj_M%2C8jH93pItAwCfwM%2C_&usg=AI4_-kRmvje-1SWbKMUjj0UGW70qBOWNyA&sa=X&ved=2ahUKEwjlqbqZ5836AhUFXXwKHfEkA7IQ_h16BAhMEAE#imgrc=uGXGdJ2vZJ2GIM