Daily bike commuter here. Wife = occasional bike commuter and rarely if it's "cold" (and I'm talking > 0 degrees in NC winter). In winter I am 1 of 2-3 cyclists in a building of 500+ people. Everyone acts like I'm Bear Grylls on wheels (that and because I take long vacations to go long-distance hiking). Many ask questions and think it's cool, but none care to try for a myriad of reasons/excuses.
My point? A person's personality plays a huge role in these decisions. I have a somewhat adventurous personality, and I like the challenge of pushing myself in less conventional situations. My wife? Not so much. I don't mind being cold in the first 5-10 mins of a ride. My wife does. I don't mind getting wet on the commute. My wife does.
We went car-light a few years ago (1 car for 2 drivers). My idea to try, but we did it on MY back because any other way would have been presumptuous. I bike commute always. She gets first dibs on the car. I even figured out how to catch 2 buses to the next town for 2 days of jury duty. I rent a car for the occasional need when we HAVE to use 2 cars at the same time.
I don't even suggest new bike commuters to try to start in the winter. Wait until spring. In your case, learn which battles are worth fighting. When you can put your butt out on the cold pavement, then do it. Maybe she'll follow. Maybe she won't. In your OP and in your follow-up, you described your wife and didn't use a single descriptor to make me think she is currently of a mind or personality to inconvenience herself over this one-mile commute. Less talking, more doing is the only chance, and that still seems like a long shot given what you've said thus far.
Good luck.