Here in the USA anyway, cellular service has been considered "critical infrastructure" for quite some time now, mostly due to the decline of landlines. 9-1-1 having high availability has been legally required for a long time, and those requirements shifted to the cellular network as people ditched their land-lines for cell phones at home. So all the towers have short-term (15+ minute) UPS's and a gas generator that auto starts, with requirements to run periodic tests.
The other part of it though is the towers nowadays require internet access to function. We had a massive storm system move through the area a few years ago with close to tornado-speed "straight-line winds" that took out a huge amount of above-ground internet infrastructure, rendering cell towers functionally disabled despite giving out full bars. There were a few lines still up but everyone's home internet was either down or spotty, and it was hard to get a cell call to connect. Was llke that for 2-3 weeks, really annoying.
So, power's not the only thing that needs to be protected to keep cellular service working.