A nightmare when she gets older? I think the nightmare has begun

I would tell Mom that under the circumstances, drop offs and pick ups must be quick. Mom is obviously not in control, and if you don't want to continue the dramas, then you must take control.

Pleasantly, but firmly, tell Mom that this misbehaviour is not appropriate for your daycare or the children that witness it and let her know your new rules for drop off/pick up.