I used to live round the corner from a private school in London (ages 5-16 or 5-18, not sure).

They were never on the tube with me, but plenty of children would be arriving at about 8:15, as I was departing to get to work for 9:00. Private school children are more likely to have a longer journey to school, therefore less likely to have a direct bus, and presumably more likely to have parents who will pay for the tube/train. (Only buses are free for children.)