The experience of DC getting 50% scholarships I mentioned is from 1-2 years ago, post VAT.
That's at several London top schools.
Public schools can offer more for the right candidate as they have hefty endowments.
OP to stand a chance for a scholarship, you basically need to do the following:
identify your DC talents early.
invest in developing these talents (tutors, lessons, clubs, coaches).
get your DC used to doing hard work literally every day (music practice, sports training, extra homework) and work hard yourself (take them to lessons, fixtures, lectures).
decide whether you would consider boarding.
make a long list of schools that may be of interest. Research them thoroughly, go to open days, speak with admissions, bursars and head of departments.
make a short list.
apply.
Your spendings on talent development will likely dwarf the scholarship so it doesn't make much sense financially, but DC will end up with a well nurtured talent in any case which is not a bad thing in life.