A few reasons that this would be true:
- The cars built before the 1980s have relatively few electronics, and those that do don't have any major systems (such as braking, ignition, fuel injection, etc.) controlled by those electronics.
- Cars built before the 1980s also contain lots more metal and would work more effectively as a Farady cage and would make the car naturally more resistant to EMP pulses. And if the EMP "bomb" was set off outside of the car and it was hardened against the attack it is much less likely to have any issues.
- If the equipment was at least temporarily turned off during the blast it would likely not be affected, or at least not stand as much of a chance of being affected.
- The type of EMP attack used will change how it affects the electronics. I haven't seen the film so I can't verify which type of EMP attack was used. See this paper as reference for HEMP attacks.
As far as the communications, this could be handled via high frequency VHF radios which were hardened against EMP pulses. Something such as hardened HAM radios or even vacuum tube based technology would prove to be resistant.
Again, I haven't seen the film, but offering possibilities of how this could happen.
Here is an article all about hardening your own gear against EMF attacks.