The article mentions cell-tower batteries, but it sounds like (in the UK) they are only intended to last a few hours to get past relatively short power-cuts. In the scenario, there was no-one left to travel around recharging them. After 24 hours with everyone dead, the cell phones would be dead too.

I'm going to do a dr test in Jan, I'll find a steaming music site and let it run and see.

Moot point, after 24 hours with everyone dead there wouldn't be anyone to call anyway

Even if most people weren't dead by then, the system would be so jammed up even text messages would be delayed by hours, if not days. I've seen what happens when an exercise involving heavy comms gets too frenetic, and the whole comms grid just locks up after a while. Then you have hazmat crews standing around in their level A gear unable to deploy, FF units pulled off to the side of the road waiting for a clear channel, and dispatchers with their heads in their hands on the console sobbing.

Text messages typically run one a separate channel than voice. Thats why in times like 9/11/2001 texting still worked. Land lines were just as jammed as the cell networks, My desk phone in my office couldn't call my wife's desk phone in her office.
My phone supports wifi (has to use a small card which I EDC with the phone) so if the phone networks were that bad I could search for wifi and e-mail messages out.