Ethernet, and the Internet, were build from the ground up to survive such a catastrophe. My best bet, although boring, would be that surviving copper pairs would be used, power generation would be very local (generators, solar, battery-operated microservers, etc.) and worldwide routing would coalesce slowly from there. We'd see random pockets of the world appear and disappear from the global connectivity since there would be little redundancy, akin to the old days of FIDOnet. With time, any dark fiber that survived would light up, redundancy would build up, and more people would have access. Wireless transmission, IMO, would take a longer time to appear since they are more power-hungry: first packet radio, and wifi only if laptops survived relatively intact (although short range would make wifi pretty useless, except in larger population areas).