Not if the crisis involves mains power outage - cell towers don't generally have backup power so mobile phones will be useless.
Edit: Adding generators/batteries to a significant fraction of cell towers is (probably) impractical due to them being bulky, expensive, and attractive to thieves.
This is not correct in the case of Norway, at least - I believe (mind, believe, I do not know for sure) that telcos are mandated by law to have emergency power supplies available; whenever we have a massive power outage, cell service stays up for hours, if not days afterwards.
Them being bulky is a major deterrent to theft, as a lot of our cellphone infrastructure is way out in the boonies and not really accessible unless you REALLY want to get to it...
It isn't the Telcos I think about in this situation.
It is my own phone batter. Or my spouse's - he has a dying phone with a dying battery, which doesn't always like to hold a charge. I'll be good for a while, sure, but once I'm dead... I'm dead. If something would happen late at night, after I've fiddled with my phone all day, I might start out with very little power. Early in the day, and I could make it last most the weekend.
Ah, that figures. Mea culpa. In that case, a $20 power bank should provide at least a couple of full charges - but if you're anything like me, you'll forget to top it up at regular intervals and will find that its capacity is greatly reduced when you need it... :-/ (I cheat; I have a small diesel generator for use during outages - we're typically without utility power a couple of days every winter.
Still, any solution based around a smartphone is going to be subpar, compared to the dirt-cheap FM receiver that can run for weeks or months on a couple of ubiquitous AA batteries.
Edit: Adding generators/batteries to a significant fraction of cell towers is (probably) impractical due to them being bulky, expensive, and attractive to thieves.