I think everyone forgets that this is set in the future... It just looks like the 50's. They are more experienced in nuke tech than we are now.
EMP damage basically causes any delicate wires to generate a charge that fries them. Imagine connecting your PC to a 3-phase electrical supply.
A lot of military tech is hardened against EMP damage even today. Plus anything built AFTER the emp has gone off wouldn't be affected. With all the fission batteries you come across (even back in Fallout 3) you would have enough juice to run anything. Even your Pip-Boy is probably run on micro nuke piles.