when there is a power outage it is most likely that the cell towers are down, too
Most power outages are local, not regional. And cellphone towers will work at a surprising distance.
Therefore my experience has been that cellphones tend to remain up, even though the power is down.
In the US, cell towers have battery backups so emergency calls can still go through. I imagine most countries do too.
Cell phone towers and communication systems have backup power for emergency communication during power outages.
If you have backup power for your router and ONT/Modem, you should also still have internet service during a power outage. The ISP-owned ONT for a place I lived had a little lead-acid battery attached to it, and during power outages I still had internet service.