On iOS you can just hold the side button and one of the volume buttons to bring up the SOS menu, if you keep holding the buttons it'll sound an alarm, do a countdown, and call the emergency services. You don't actually need to interact with the screen. Obviously this means you'll need to be able to squeeze your thumb and another finger together, but a phone with buttons would require you to be able to operate that somehow too.
I think you could also try to ask Siri if that's enabled.
I've no idea about Android but I'd assume you can do something similar there.
It does on iOS as well. Though note the difference between holding and pressing. Pressing Power + Volume Up takes a screenshot, holding it (keeping it pressed) brings up the SOS stuff.
holding the power button brings up the emergency menu. You can also use the same menu to lock your phone down so none of the police scanner software works.
And I'm sure you always keep it with you as well. Really, good for you. Amazing. So many people. And I happened to just comment to the one person who keeps a dumb-phone on them. What are the odds