We had a (much smaller) regional (Lübeck and Ostholstein) power outage here in Northern Germany in I think 2018 or 19. No mobile internet until everything was restored ~6 hours later, and most of the time no other cell service either.
Always picture someone with a big can of diesel filling up a generator. Experience says it could be a genny running on natural gas. Had that at work before.
I think cell phone towers use surprisingly little energy, just a few kW. So even longer term operation should be possible withs bobs backyard generator running on vodka.
But I would guess the whole network equipment would draw quite a bit, especially a modern infrastructure.