I think people who want private education for their children are typically well-off parents who can also afford to pick the more affluent suburbs/dormitory towns to live in eg Harborne, Edgbaston, Solihull. They are often prepared to do the trade off of long commutes and traffic jams.
Both car traffic and public transport round the West Midlands is frustrating at peak times. Journeys of a few miles can take up to an hour, particularly if you have to change buses. (A lot of the main routes go in and out of the city. Buses that go around the city often take all sorts of diversions to scoop up people from housing estates. It is honestly easier to walk 3 miles than take a bus in many cases.
So, if you're on a budget and not fussed about living in a posh area, I'd go for a less upmarket area on a direct bus route - ideally where you'd be able to meet up with other HK expats. There may be fewer other children going to the same school in the locality, but as KEHS does offer bursaries and scholarships not every child will be based in the poshest parts of Birmingham.
Also there are many less expensive areas that nonetheless have access to good shops, and which are near green space. It's worth saying though, that Birmingham council has - effectively - gone bankrupt, and so some public services will be cut causing problems in the city. But a lot of local authorities are in trouble. I can't imagine the situation is significantly different in Manchester and a change of government may result in better investment in the regions.