[imglorp]

Some of the cellphone network can operate with no wired power or network. Towers can have p2p microwave links and battery backups. That doesn't really answer your question about how/if/how-long it worked this week.

http://ups-battery-backup.com/cell-phone-ups.php

https://enterprise.spectrum.com/solutions/carrier-services/c...

[jpindar]

Some of the ones near me even have small wind generators.

[TomMarius]

These probably were super-effective during this occasion