YANBU.
I have had the same nanny for over three years. As you would imagine, we have a good relationship. She is not my friend, but we can tell each other the truth, I've met most of her relatives and taken her abroad.
I cannot imagine any circumstance where anyone would ever ask me for her phone number.
I can, at a push, imagine our next door neighbours, who are good friends, and who know our nanny, might ask for our nanny's help if they had a medical emergency. However, this has never happened (even though they have had medical emergencies) and I am sure they would tell me their circumstances, and ask for my permission, first.
I know that my nanny has, on rare occasions, looked after other children during the day, for instance when her nanny friends needed to go the gp. As far as I know this is an exceptional occurrence and I have chosen to close my eyes on it as I know these nannies personally and I want them, and our nanny, happy and healthy. My oldest is now big enough that she would be able to tell me if this happened, so I can't imagine my nanny trying to do this behind my back.
All of our friends know we have a nanny. Many of them have met her several times. No one has ever asked for her phone number. I am 100% sure they would never do that without checking with us first.
That mother is a cheeky mare. I think I would either ignore her text, or call and ask what it is about. If she said that she wants our nanny to pick up her kid at school, I would very quietly tell her that I would expect her to check with me first, as I am the employer. And then I would ask her how much she is proposing to offer my nanny for the extra work, so I can advise her whether this is a good opportunity or not.