I started painting my granddaughter's nails whilst she was sitting on her potty! She was always very interested in nail polish. My GD told me firmly (aged 2 1/2) to paint her nails "properly" - she wanted base coat :lol but be warned, my GD chews her polish off. So you do have to think about product ingredients. Also acetone free polish is too aggressive for young skin and can create a burning sensation (ask me how I know this...)
I occasionally paint the nails of an accompanying child in my salon, but I never charge for it. It's actually terribly difficult to do a professional standard job, children just can't remember to sit still and not touch (their hair, their nose, your furniture)
I've had a client bring in a 12 year old for a mother and daughter mani and pedi. I'd not have accepted the booking if I'd known but it was the child's birthday so we could hardly turn her away. The child looked thoroughly bored and miserable. Whatever she had imagined, the reality wasn't it. We used the diabetic client protocol on her.
I've also accepted a booking for 14 year olds accompanied by a parent. The parent was thrilled but some of the girls had really terrible cuticles and nails so the end result was very disappointing.
As a general comment, salons are not safe environments for children. I hate it when I have kids in salon because it's really stressful.