Get mine cut in Mercato in the Fun City hairdressers for kids. Kids DVds, high chair. Only does kids. He'll take your name and call you when he is ready. Opening hours a mystery but we book our place on a Friday or Saturday late morning and go for lunch in one of the food places there. They have the same thing in Ibn Batuta Mall but the queue is always longer!
MoE and Dubai Mall do not have a kids hairdresser to my knowledge.