The Internet at large is designed to resist nuclear blasts. At least, it was a design goal of its immediate predecessor, ARPANET.
There is no secret: to survive loss of components, you must have redundancy. In the context of nuclear blasts, this means that there must exist several paths for data between any two machines, and the paths should be as geographically separate as possible. Mathematically, given an assumed blast radius r of 50 miles (for a nuclear-powered EMP, this is a rather low estimate), and two machines A and B and two paths between A and B, then the following should hold: for any two points M and N where M is on path 1 and N is on path 2, and the distance between M and N is less than r, then either M or N (or both) is no farther than r from either A or B. In plain words, the two paths never come closer than 50 miles from each other, except at both extremities (the two paths necessarily join at A and B).
The packet routing nature of ARPANET, then the Internet, allows for such redundancy. Extra points to radio links, in particular satellites: the link between a base station and a satellite cannot be permanently disrupted by a nuclear blast in between. The blast may induce a high ionization of the upper layers of atmosphere, so communications may be temporarily jammed, especially for longer wave lengths; satellites work in the GHz band and should have less trouble than, say, FM. Also, geostationary satellites tend to be relatively high on the horizon (at least from southern USA -- much less so from, say, Moscow) so getting a blast between a base station in Atlanta and a geostationary satellite which is roughly over the Americas entails detonating the thing over US territory, at which point Atlanta itself is in big trouble.
Transoceanic cables should also be fine: 3 miles of water are a Hell of a shield. And they offer lower latency than geostationary satellites (the ping time with a remote server through a geostationary satellite cannot be lower than half a second, because 4*36000 = 144000 km); latency is a problem for flying drones. Lower-altitude satellites are more difficult to use (from the point of view of a base station, they move a lot and often go beyond the horizon) and are in range of Anti-satellite missiles.
Optic fiber is more resilient to EMP than copper links, and the US military have studied that for more than 35 years. The weak part of an optic fiber link would be repeaters: the devices which pick up the signal and re-emit it stronger. You need some of these in any long-range cable. But at least this reduces the problem to building anti-radiation bunkers at regular intervals.
Actually, a bigger problem may be electricity. An EMP will imply high surges in the grid. For instance, the US grid has trouble resisting bad weather.
And, of course, redundancy of network links is not sufficient: you also need to duplicate the servers (data storage, computing elements). You already need to do that to survive floods and earthquakes and even simpler events like a server room burning down. EMP resistance is just more of the same, on a slightly larger scale.