What types of information should be treated confidentially in a nursery setting?

Names
Addresses
Phone numbers
Health issues
Medications used
Behavior issues
Academics or abilities

Always wise to use the approach of whether it is something that is necessary for that person to know. For example... don't give a parent other families' addresses for something as innocent as sending out birthday invitations. Be careful when discussing a student's behavior or abilities. While the teacher and student's parent would obviously need this information, another parent nearby would not.
If there were any allergy concerns, for example maybe a peanut allergy, the person who prepares meals would need this information, as would any staff who might interact with this child. Parents who might be bringing in snacks or birthday treats don't need to know the child by name, simply that one of the students has a peanut allergy.