Amazing/curious that all the infrastructure to feed news to your phone still works. I can imagine a poorly-built systems would have some infrastructure without UPSes, and in all that chain of technology, there'll be an element or more without power. Not saying Spain/Portugal would necessarily be more prone to have poorly-built infrastructure, I assume the whole world runs on half-assery.
If their networks are built like the ones I've worked on, everything has at least a little battery, but only some of it has generators to keep going after that.
So some or most cellular towers will have generators, and their fibers will backhaul through repeaters, some or most of which will have their own generators. When it gets back to the MTSO, that will definitely have large diesel turbines on site and at least 24 hours of fuel with priority refueling contracts.
I'd expect there to be a lot of outages, for instance where all the towers in a region end up backhauled through a site where the generator fails or was never installed for some reason. But there will also be a lot of places that stay up in some capacity because, more or less by happenstance, all the fuel tank permits got approved and all the equipment actually worked.