FWIW, I looked at what would be needed for private schooling where I live. I realized socareless forgot to add in things like clothes - which can add a bit to your costs.
Expenses per month
Rent for £1,200 for a 3 bed semi
Utilities £250
Council Tax £160
Food £500
Travel £500
Contingency/savings £300
School Fees for Prep £1,100 (for one child)
Clothes £130 (quite modest for a family of three!!!)
That's £4,140 net per month that is needed. Plus, I do think parents should pay to a pension pot, and there are things to be paid like uniforms, car tax, insurance, etc. so we will roughly put this up to £4,500 per month.
So if both parents work, and get equal wages, they EACH need to earn about £36,000, giving a total household income of £72,000.
If you want to have a more comfortable life with one child only who goes to private school PLUS holidays, extra cash, kids having toys, etc. I think you need a household income of at least £80,000.