@ OP, YABU and, dare I say it, sound a tiny bit jealous, petty and resentful
Firstly, you have no idea if the parents "bully" their children - what is bullying to you is probably simply firm parenting to them.
Secondly, to imply that the children "bully" their nanny - frankly, the nanny sounds as though she is in the wrong profession or simply not as firm with the children as she should be (or as the parents have directed her to be). After all, if your child was to throw a tantrum around you, you wouldn't necessarily say that he/she was "bullying" you, would you?
Thirdly, unless you live with this family 24/7, your point about them never smiling is most likely inaccurate and ill-informed. I'm guessing you only see this family in a formal social context i.e. at the school, not in their front living room on a daily basis.
Fourthly, if their DD is going to stay with a lawyer colleague of the father, I'm guessing the household she will be going to will be fairly affluent, comfortable and (given that she will be a guest) relaxed for her. She probably won't have to lift a finger in the form of household chores and will most likely be personally chauffeured around to be shown the sights.
I'm guessing it will prove a great adventure and holiday for her and invaluable when it comes to life experience, and hardly counts as abuse. All because you obviously don't hold your own child to the same impossibly high standard or give her as much freedom does not mean that the other lady is abusing her daughter.
PS - "Pushy" is usually used to describe people where one feels inadequate and unable to emulate the high standards shown by and expected of others. This usually results in the latter forcefully convincing themselves that they would never wish to emulate those standards anyway and condemning those who choose otherwise, as opposed to admitting to their own shortcomings and feelings of inadequacy.
The long and short of it is - YABU.