Why is the speaker using "not before" instead of "after"?
Answer from a Native speaker
Rachel uses "not before" to add on to her previous statement, "Right after you left, they wheeled her off to the delivery." Using "not before" here shows that this event happened before she went to the delivery room. "Not before" can also be used to indicate that you want something done before doing something else. Ex: After breakfast, you can go outside, but not before the dishes are washed. Ex: She left the room crying, but not before she cursed the teacher out.