Nanny's catchphrase of "When you don't want me, but need me, I will stay. If you want me but don't need me, I will have to go." In the first movie she even lampshades this as she finishes, noting how tragic it was. True to her word, her departure is a sad one.
What makes it even sadder is that, in the first movie, she just quietly slips out while everyone else is busy with Mr. Brown's wedding to Evangeline. No one gets a chance to say goodbye to her.
The second movie invokes even more feelings. As Nanny McPhee is leaving, even the mother notes how they all still very much needs her. As she turns the corner, the mother sees Nanny McPhee on the horizon and realizes they really didn't need her anymore; her husband has returned home from the war, injured but very much alive (they had thought he died). Nanny McPhee remarks that her final lesson was complete; To have Faith.
In the first movie, when Mr. Brown finds out that someone DID go with Aunt Adelaide, and presumes it to be his beloved daughter Chrissie. Frantic, he chases after Adelaide's stagecoach but is unable to catch up. As the stagecoach rides away he wails the loss of his daughter... when suddenly all the children catch up to him, including Chrissie. So... who left with Adelaide? Evangeline, the scullery-maid. Also a Crowning Moment of Awesome for Evangeline.
After the children drive Mrs. Quickly away, there's this moment where Mr. Brown just looks so defeated as he tells them that she was the last thing standing between them and financial ruin. Them learning exactly how bad the family's situation is prompts them to make amends and convince her to marry their father.