My 9 year old daughter (Year 5) goes to Inspire Ballet and Dance in Balham. In the church hall opposite Waitrose car park. She loves it.
I would imagine it would be possible to do private lessons as you suggest to speed her along.
Some of the girls in my daughters academic year took Grade 2 last year. Due to COVID it’s obviously been tricky to do exams so her class haven’t managed to do the exam (they probably should have done it in the autumn) so I’m sure there will be catching up to do for lots of them!