London is pretty big, depends where you want to stay. Some independent day schools don’t offer CAS so it would be difficult for your kids to apple for Visa.
In terms of prices. For Secondary school, it ranges from around £15k to £26k p. a. for London; SouthEast/ South West around £18k to £22k; Manchester, I only looked at Manchester Grammar School 2 years ago and it was £12k
Subscribe the Good School Guide that will give you some ideas