The key question here: How close are you to the mom? Personally? Is she a dear friend with whom you would feel you could talk about this? If so, talk t to her (but NOT with any children present -- in fact, I'd take her out to lunch or for coffee without kids somehow because it sounds like she needs it). "Sally, your daughter is such a sweetheart with me and she's easy to watch. She respects my rules at home and listens to me. And I value that. I also value YOU as a person, and I know you're more than just a mom. So I wanted to tell you that I'm really troubled because while she's great during her time at our house, at pickups I feel like she suddenly changes -- there seems to be a lack of respect for you, and fussing to get what she wants. How can you and I work on that, so that the pickups are not just easier for her but for YOU? I care about you both." Make it about the mom and having her life be better. Praise the girl for what truly deserves praise. And focus for now just on the one concrete thing -- pickup time issues. Don't get into the girl's general manipulativeness and the fact mom caves in to her and clearly doesn't discipline -- you are out to help and support mom, and talking about other issues, though they so clearly exist, would make a depressed mom worse.
Before the talk, have some suggestions in mind, perhaps ideas about how you can give the kid a 15-minute, then 5-miinute heads-up: "Your mom will pick you up in 15 minutes..." "In five minutes...." "I expect you to get all the way to the car and into the car with both you and your mom smiling; if you can do that for me, (reward)." Anything concrete.
Is the mom in any counseling or therapy that you know of? It sounds as if she could use it if she is depressed (diagnosed or just your guess) and she also sounds very overwhelmed. That's why she gives in every time -- she is mentally exhausted by this child and giving in is easier. My heart breaks for her. If you are really close maybe this discussion can lead into your asking her if she is getting help or maybe even interested in seeing a counselor to talk about her daughter and herself.
I can say that letting her child go on like this will yield a child like my niece, who can be a total delight but who has grown up ruling the roost (despite her parents not being depressed, having two parents in the home, etc.). The result is a child who is so self-centered and rude to adults and other children. She's a popular girl with peers because she's very vivacious and funny, but when she decides she's bored with you, or wants something -- watch out. And she's 12 now, so the pattern unfortunately seems set.