Is it possible for my baby to become too attached to her nanny?

No. Attachment is a good thing. It will be easier to transfer secure attachment to you, the parent, than try to achieve it when the child never knew it.
No. Your baby being very attached to your nanny is a positive thing. If you are a loving, caring mother you will always be #1 in your baby's eyes and your nanny will be done on the list.