Can the Cellular Applewatch operate without an iPhone? My 80 y.o. mother will not use a smartphone, however, I would like to give her a cellular (T-mobile service) Apple Watch, because it sports the fall sensor, that automatically calls EMS / designated contact.

Although could configure the Apple Watch with an un-simmed iPhone:  Is it possible to provision an A-Watch so that it has its own phone number and not share it with an iPhone?  She has a flip-phone (Tracphone) that she is comfortable with and I am certain she will switch to any smartphone.
My suggestion: Buy a cheap refurbished iPhone, bond the watch to it, then put the phone in a drawer. The LTE Apple Watch doesn’t need the iPhone to be on or nearby if it’s a Series 4 or later. It is NOT possible to set up an Apple Watch on a number that s completely separate from its iPhone bond mate.

BTW, she can make calls from the Apple Watch without the iPhone involved, but it has to be bonded to an iPhone.

I’m betting you know somebody who has an iPhone 6/7 with a shattered screen or bad battery.