I agree with Metella. In a sense the private schools are an irrelevance as are the home educators. Virtually all children are in state schools and some are better than others. We used to cream off the clever children to grammar schools and that used to work for those creamed off but most areas don't have grammar schools now. Education, education, education of Labour has ensured a lot of money has been ploughed in including a lot of private mnoey into new things I cannot evn remebmer the names of like city academdies and schools with ridiculous claims to being a sports or music specialist when in fact their standards of sport or music are way way down and much much worse than any state grammar or private schools and it's a con. If it's a specialist music school then get them doing proper music. Compulsory choir with latin difficult parts to sing. Force every 11 or 12 year old to do grade 5 music theory. Make them take associated board music exams. If it's sports have the morning run every day for every one, get them to the top of the leagues including against private schools - be excellent.
But all those places are quite divisive and in a way simply perpetuate the errors of the grammar/secondary modern/ divide.
I'm not sure if I would want total uniformity though in schools. Parents should be able to pick Summer Hill (no obligation to attend lessons), Eton, the Purcell School, Manchester Grammar, or a very strict religious school but perhaps they shoudl noly get the choice if they pay and if my money is instead being used then because we need educated children we need to enforce more learning.
What do we need then? A school cert at 16 where you must do separate English li and english lang, maths, French, Geography, History, biology, physics, chemistry or two of those sciences? That's fine even for children with a low IQ as they just get low marks in it and plenty could fail.
Rules - making them stand up when the teacher comes in the room, calling male teachers Sir, strict uniform policies, removal of disruptive chidlren to special schools very quickly; single sex education; probably lots of other things too