If you're making her eat after she's full then you're teaching her ignore that feeling and to over-eat. That's NOT a good thing. You're putting too much food in front of her. Her stomach is only the size of her fist and therefore that's the amount of food you should place in front of her per meal.
You need to give her smaller servings and if she wants to snack in between meals, then let her. When she feels full, let her step away. It means she's had enough. You don't need to force her to drink before a meal or after. Let her drink in between meals or small amounts during the meals. She needs to be able to follow her own hunger and thirst cues.
The nanny is making her drink "a lot" of water? She should have water available to your daughter so that when she's thirsty it's there for the having. But don't "make" her drink. Just like having snacks available for when she asks.
Feeding her while she's playing, just spooning yogurt or whatever into her mouth, is teaching her to eat when she's not hungry as well. It's teaching her that she can't even perform a leisure activity or sport or anything without "needing" to eat. No! Don't do that! If she's playing and is hungry, she can take a break and eat where she's supposed to.
I'm assuming here that your daughter isn't underweight; isn't off the charts underweight or in the very low percentiles and is not considered to be "not thriving." If your child is thriving by medical standards you and the nanny are overfeeding her and teaching her to overfeed herself when she's older. Setting her up for a life of obesity.