How do we talk to our mother about not driving anymore without upsetting her?

If she is a danger to others don't be so concerned about upsetting her. Talk straight to her. Ask her how she would feel if she had and accident and killed a child or left a child without parents.
When Dad wouldn't stop driving his car keys just got lost one day and were never found. Mother had the good sense to stop driving on her own and gave the car away so she wouldn't be tempted.

This is so hard. Giving up the keys means you have lost all your freedom, and now must depend on others. It is heartbreaking. My Dad has parkinson's and would freeze for seconds at a time. Nobody in my family would deal with it. Finally Dad and I had a talk, He knew he should stop driving he just couldn't get his head around it. I flat out told him. He is a danger to everyone on the road, because he can't get his foot from the gas to the brake fast enough to avoid hitting people. I looked him right in the eye, and told him I would call the police to report a dangerous driver on the road. Before he left the driveway, they would be here. They would start proceedings to take his license. I meant it. He knew I meant. He stopped, after a year he sold his car. He still gives me the evil eye every once in awhile. I'd rather have him angry at me, then the guilt of a person's injury or death on my conscience because I was afraid of upsetting him.

Unfortunately, my mother thought she was good enough at driving to run around with her grandchildren. I rode in her car to test her skills and she ran into a ditch, drove through a red light and ran a stop sign. She called me a liar when I confronted her about it.
My sister and I are guardians when we talked to her about our concerns, she said she was just fine - we didn't agree. We drove off in her car and she never saw it again. At this stage of her dementia, she didn't comprehend how dangerous she was and we were afraid, as guardians, that we would be held liable if she hurt somebody or damaged other people's property.
The car had to be removed, not disabled, she would have had someone come fix it and there would still be a problem.

My return button got hit before I was done. There's no talking about this IMO because if they/she could reason properly or at all they would be scared to death knowing they are getting lost. Take action first, remove the vehicle and the keys and discuss later. If you have to find a fall guy (if it cannot be you) let it be her doctor, pastor, police officer etc. Reporting her to the DMV via the doctor could result in a formal letter from the DMV but it doesn't stop her from operating the vehicle. Even with authority figures terminating their driving rights, some people will ingore warnings/laws and drive anyway because they do not/can not reason. You have to do it for her before someone gets hurt.

This resolution is more cut and dried, black and white if you will, then some other issues. This isn't just about the feelings of our elders, this is about death or serious injury to our elders or others. This IS NOT EVER a decision that can be left up to them because their reasoning ability is not always intact. The approach to reason with them, for example if they are safe driving their grandchildren (and as we just read, some thought they were when they weren't), only works if they ARE able to reason and their reasoning is reasonable. If their reasoning is out to lunch in any way, then "the disappearing keys" gambit is the way to go. Then redirect, redirect, redirect to get around the where are my keys or where is my car business. When it is no longer their time to drive, they just CAN'T be permitted to, period. Alliwing them to continue to drive, even in order to preserve they're feeling of independence, is just too dangerous.

Can her doctor make a referral for her to take a driver's evaluation? Maybe a neutral party, such as a TRAINED occupational therapist can get some hard data and give a scientific assessment that would remove all emotion from what is truly happening. It sometimes helps to have a third party deliver the news. After all, you are just her child.

She may not be as upset, if you can, at the same time you tell her she cannot drive anymore, propose another way for her to keep her independence such as having in place a car or taxi service or hiring a college student to take her on errands.

Let's see, danger to herself and others or her independence.

We got the MD to refer her for a road test. She passed the road test, we let her drive until this May, then we borrowed the car and never brought it back. It takes a neurologist to tell her she cannot drive, and she denies he ever said that. Take the car and stand your ground.

When my mom was first diagnosed with early stages of dementia, she asked me about whether or not she should drive. I asked her if she felt she was a good enough driver to transport grandchildren. The next morning, she announced that she would no longer drive and asked what she should do with her car, which we helped her sell immediately.