Yeah, I think they shouldn't have called them cell phones, they looked like just AM radios, which aren't that complicated. Just need a microphone (membrane attached to a magnet), amplifier (they used vacuum tubes), and an antenna (whose length would determine the frequency). Receiver is the same thing but backwards (with adjusted geometry on the mic/speaker, but they are electrically the same). If they had the vacuum tubes, copper wire, and magnets (which they could make more of once they had one), then I don't think that part was that far fetched (though I could be missing something big tbf). But a cell phone is a lot more than a 2-way AM radio, even if you're talking about a simple phone with no display.

Biggest issue I had with suspension of disbelief was when they used a vehicle without a road system. Roads came before cars in our history and had to be good enough for horses to pull carts. Using a two stroke steam engine to generate enough torque for offroading with those basic tires was a big hand wave IMO.

Oh also the whole using a celebrity impersonation to convince some of the other side that the US was back up and running was a weak tactic. Like I'm glad they had it immediately get noticed, but a better way to do that bluff would have been to just have other "radio chatter" going on to make it sound like they had joined up with others, at which point they could have talked about all kinds of technologies that might have taken the fight out of people from the modern era.