Personally, I don't think you are being nosy.
You are concerned for the welfare of this young child who is 10 years old.
Obviously you would need her father's approval to either make her an appointment for you to take her to see the doctor or for the father to take her. Write down your concerns on a piece of paper so that the father can give it to the doctor as dad's don't always pick up on emotional issues with their children.
I agree that her nutrition is not very good. But the main issue is to help her to deal with her grief for the loss of her mother.
The doctor would hopefully refer her to counselling that specialises with child bereavement. Once the mental anguish and loss is dealt with, I am sure that the pulling out of her hair and pulling up of her skin will stop.
When you are with her, talk about her mother. Just because her mother has passed away does not mean that she should be forgotten.
Young children sometimes blames themselves, so do reassure her not to blame herself for anything that has happened and that her mother is now in Heaven.
The father and sister may benefit from bereavement counselling too.
Everyone's grief is expressed differently and it can take many years for them to come to terms with their loss. Unfortunately, some people never do get over their loss and are unable to move on and live their lives.
If she was taught to say prayers, when you are with her, say a prayer with her to her mum. If that is not your thing, perhaps you could still do that. Anything that help the child is worth doing.