It's interesting comparing the governance of schools in the UK and Australia (or, more precisely, in England and New South Wales). The headline figures are that only 7% of children in England attend private schools whereas more than 30% of children in Australia do so. But examination of the details makes the difference much less: many state schools in England seem closer to me to Australian private schools than to Australian state schools.
Government schools in NSW are run by the Department of Education. Local government doesn't get a look in, churches don't, and there's no equivalent of parent governors. There is some variety in schools — selective, specialist, rural multi-age, and so forth — but they take teaching staff from the same pool and draw on the same resources. School exams are set and administered by the NSW Education Standards Authority across the system (also for private schools, unless they opt for something like the International Baccalaureate).
Private schools in England are, though they come in radically different forms, also quite simple formally. They receive no government funding at all and are subject to very light government regulation and oversight (this also holds for homeschooling). And the A-level exam system allows a choice between different (all private) exam boards.
In Australia, private schools can be 90+% state-funded, some get more per-pupil than state schools, and even the very richest private schools get considerable state funding (whereas the idea of funding Eton and Harrow is almost unthinkable in the UK). In NSW there is an entire Catholic shadow education system that still attempts to provide comprehensive coverage.
Government schools in England are much more varied, largely because the state itself does so little for them other than providing funding. They get some support from government if they are local authority affiliated, but not much. And the actual governance of schools can vary quite a bit: some are responsible to local authorities, some are church foundations, some are part of academy trusts, some are free schools. Coming from NSW, an English school run by a private academy trust seems to me to be closer to a state-funded private school than a state school.
The governing body of an English state school might consist of local authority or church-appointed members, the head teacher and some other members of staff, and enough elected and co-opted parents and outsiders to get to ten or twelve. These are unpaid and often untrained volunteers, but they are responsible for overseeing school finances, writing advertisements when a new head teacher is being hired, interviewing them, and so forth. One result of this is even bigger geographical stratification than in Australia. A state school in Sydney's North Shore draws on a completely different student demographic to one in its outer Western Suburbs, but they are part of the same system. In England, school governance varies much more. In more affluent areas, schools can find businessmen, accountants, academics, etc. prepared to serve as school governors; in poorer areas, this is not so easy. Using volunteers may produce excellent results in some cases, but it can also produce poor ones, and tends to exacerbate social and geographical inequalities.