E4B takes what? 100 people, they'll probably go for commonality with 747-8, probably the freighter or a ULR 777 or A350. A350 is unlikely because it's not American. So maybe the choice is 747-8 freighter, 777-8 or maybe a 787 on steroids. I don't see the 767 happening. Sure most of the computing power needed today could be done a couple, ok a few large PC's but the E4 needs to work in a nuclear exchange and digital electronics have a habit and not a practically fixable one of going awry under increased radiation levels etc... even more so on top of the line modern PC's whose brains work at or close to the atomic level, which is a bit shit when you're going to be bombarded by a lot of alpha particles in your mission... survive and control during a nuclear holocaust. the USAF will need the size of a 777/747 for the semi digital/analogue computers currently in use which WORK and probably very well (otherwise they wouldn't use them on their last ditch effort to command their armies), sure they'll be updated and better blah blah but you get what I mean. I don't see USAF using a 767 or even a 737 for that matter. That's my opinion on this E4 replacement....
You mention worries about computing in a bad environment like that. What do you think would happen to all the electronics on the B77X or B787 in such a situation? You'd have to add tons of equipment and shielding to deal with such things. And it'd be interesting to find out how easy it would be to make a B787 a faraday cage with the composite fuselage. Finally alpha particles are stopped by a thin sheet of paper and even beta particles are stopped by a thin sheet of aluminium. It's gamma you'll be worried about. And anything capable of taking out electronics with enough energy to destroy them will likely be a pretty quick kill for anyone in the aircraft.
The B767-2C is the front runner as it's in active production and it's big. Not 747 big, but it'll be big enough. And it already has had all the military requirement tweaks done. Anything specific to an E4 replacement is the only mods to be made.
If the USN can make do with the https://en.wikipedia.org/wiki/Boeing_E-6_Mercury
then the US Sec Def can make do with something bigger but maybe not quite 747 sized.