So Strike 1.
Tell Grandpa thanks but to relax. And tell her to step it up.

No OP wants her to arrive at 6 because that's when she leaves. Grandpa's not to be responsible for the kids. Nanny fell asleep missing getting the kids up at 7 . Grandpa had to wake her.
The more I think of this the more I'm bothered by it.
This is the nanny's third day on the job and nothing unreasonable has been asked of her.
If she has an early start time it's up to her to get to bed the night before or get herself a coffee.

If op is OK with her sleeping till the kids wake it's on her to set the alarm

For her to be sleeping so heavily that grandpa had to get her that she missed wake up time by nearly an hour is not OK.

This was only her third day in this isn't an established nanny who has had time to prove herself.
Op is right to be concerned and contemplating letting her go.
I don't think she would be wrong to fire her over this.
If she does keep her on she needs to have a very serious talk with her. As in this is her warning and one more misstep she's done.

does she come that early to beat the traffic?

Why make 6am a start time when there is a grown up in the house and the kids need a nanny to wake them up at 7 (and they will sleep longer if not awaken by nanny). She can watch tv during that hour that means you didn't give her any special tasks for that hour. Why don't you have her come in at 7?

Just tell her to set an alarm or have the nursery came noise on high so if the sleeping kids need something or are up, she gets up!
It won't be as cute when the kids are missing 8am preschool or whatever which you will also pay through the nose for.

An aside, I know my nanny takes a nap likely daily during some/all of the child's afternoon nap. It doesn't bother me unless she is neglecting the child, or the house is a mess, or they miss a class.