Appalling that you would even consider bringing this up to your nanny. Beside the highly uncomfortable position you'd put her in, worrying about her car, she'd also be worrying if grandpa would be back before her off duty time. I know this would happen to me. I need to leave at 6pm, grandpa would show up at 6:27 and a smug face.
I am sorry that your nanny was rude on her answer but it looks like you try to skate by life anytime you can. Do you send your children on a "playdate" on sundays when you dont have your nanny so that you can go "run some errands"???? Let your dad use YOUR car!!!

While I agree that the words "Absolutely Not" are indeed a very rude response, I think you crossed the line by asking your nanny this favor.

Your relationship is a professional one first and foremost, but also you have asked her a HUGE favor. Her car is her bread and butter OP. When if your dad wrecked her car, then what would you guys do? Plus, it would damage your relationship with each other and I am sure you do not want that hanging over your heads.

Why not just have your Dad rent a vehicle while he is here and keep your nanny out of it?

It is best not to involve her anyway since this really is not her issue or has anything to do with her.

I would be shocked if my family asked this of me.

My nanny wouldn't let me use her silk blouse as a hanky to blow my nose. She wasn't wearing it! I fired her.

Geez people, it's a troll.