Mom will be aware that child needs to socialize and that is why she has her enrolled in the other program too. As far as being with the nanny - one of the downsides is that often they are expected to only do childcare and not housework so are there just to entertain the child - good to a point but damaging in the end. I wouldn't say anything to mom at all for at least a month. Give the child time to figure out her place in your group - will she be the oldest and the leader, the youngest and the follower, or somewhere in between - which kids she feels more in tune with - this will all take time for her.

I try to do a lot of group activities to show them how to interact and then back off and leave them to play together. She may just need a push.