We just moved here but in our previous city we had to let go of our nanny after catching her in lies about where she was during the day with our child.
Our child was missing her regular naps, meals and even classes because our nanny "met a mom in the park who wanted to pay for babysitting playdates".

We were never told, but figured it out quite quickly, gave the nanny a warning about her priorities, she chose to lie about whereabouts and get belligerent. The trust was broken and we got another nanny within 2 weeks.

OP - do not nonchalantly pay someone else's nanny to babysit or drive you, your nanny or your child. Bad idea. Do you know the other family? Is there a mutual understanding? Otherwise, there is often a breach of contract and a real conflict of interest is created.

In our case, our nanny started lying to us and the quality of care went down. We saw this via our nursery camera (why is our child gone for 4+ hours during naptime?) and in our whiney, overtired toddler whose schedule was dictated by "the other mom's kid's schedule."

Be thoughtful how you break this to your nanny. You hired her knowing she did not drive. It is not fair to turn around and fire her, as other posters are suggesting, when you made that call. Also, the driving nanny has little incentive to help you if you only offer her gas and mileage. She will be in no better position but have double the headache. Are you sure these nannies have a positive relationship and would be ok spending every day together?

You need to ask the nanny first. I would be pissed if my employer made the decision for me. She interviewed for the particular job they offered, it is her right to turn you down. What if her employer thinks it is a great idea but she doesn't want to do it? Better to have her weigh in first then only move forward if she is willing. Of course, you must ask the other NF but I disagree that you go to them first.