Been there, done that. I think we used to budget a grand a term .Luckily we only had all three at school at the same time for one year.There does seem to be this idea that CEA/BSA covers all the costs and you can send your child to Eton/Harrow?Rugby/Etc.
They still have to pass the common entry tests, so you might not get your first choice and boarding doesn't suit all children.
But lets say you you get your 2 kids into a fairly average boarding school. That's still about £1000-£1200 per month. plus initial uniform and kit out, budget £1k per child. Now start adding the inevitable extras and these schools bill for all sorts of things. And the school trips can be eye watering, even just going to the British Museum in London etc. Once they start heading off to Iceland, Ghana, Washington, etc.
Fun fact, even when to comes to exams get ready to pay c£60 per subject exam entry.