What "legal action"? Call the cops? Probably not. Get a restraining order? Probably not.
I am confused. Is your mother going to barge into your girlfriend's home and rip the baby out of her arms? If so, then your girlfriend needs to call the police, because that would be kidnapping. Your mother cannot pick up the child unless she has permission from the parent with whom the child is currently living.
The child is only about three weeks old. I suspect that the state just hasn't gotten around to that yet.Have you been ordered to pay child support? If so, are you current on your payments? If not, then I find your story questionable, because the state would almost certainly be seeking a child support order to cover the cost of housing your child.