Take it one person at a time. Acknowledge all of the customers and assure them that you'll be right with them.
Remind calm, ask them if they can wait just for a bit til you finished doing business with the customer. One by one helping them out and understand taking time and patience. learning their time very valuable that I take this seriously to make sure everyone gets out calmly and happy.
I would be as empathetic as possible, and reassure her that it is natural to be anxious about being away from her child. I would encourage her to watch her child's interaction in the class environment for a few minutes from the window outside the door, so she isn't directly in the classroom but gaining comfort with how her child is being involved in the class, and meeting peers. I would recommend to her to try this and communicate how she feels about it all over the course of a couple days (even a week), and to keep open lines of dialogue.
I understand that each colleague has unique value that no one else brings to the table and if they are treated with respect and support, the fruit of that value can be immense and can ultimately contribute to a successful center.