hi Sandunes, you seem to have been left hanging!
I am afraid I can't advise you about private schools, but would say that many small private schools ,which the majority in East London are , are often not able,or willing, to support children with SEN for a number of reasons, eg other parents reactions, facilities, expertise, financial cost etc. For example you will not find one able to offer 1-1for your child, if you think about that situation from their point of view they would not make any money if they were having to fund a full time TA for your child!
My best advice is to approach your local state primaries, Newham does have an inclusion policy, which means that most schools have well qualified SENCOs who have a wide knowledge of SEN. They are well funded (relatively) so can afford a good level of TA support, though again I doubt if your child would have full time support, but any support would likely be from someone with decent training and understanding of your child's diagnosis.I would call a few local schools, ask to visit, explain your situation and see what their response is.
Is your child currently in nursery? Have you spoken to them about local provision? Are you in touch with other local parents of children with autism? Their knowledge and experience could be very valuable. good luck.