Oh dear! That's simply naive and wrong. Perhaps that is typical of you?
My late father was an East End Grammar School boy in the 1930s who went on to be knighted. His school mates went on to win a Nobel Prize, went on to be much-published writers, went on to be scions of the Communist Party, went on to be long-standing Treasury ministers ...
He sent me to a minor Public School, one of whose old boys was the Labour Cabinet Minister who famously said "If it's the last thing I do, I'm going to destroy every f*****g
grammar school in England. And Wales and Northern Ireland" and tried to! Because the Labour Party has never really wanted to diminish its client base through social mobility.
There is a glorious irony in your post though, because the greatest single diminution of the British class system did indeed come after World War II as part of a response to the
National Socialist government in Germany in the 1940s, But I bet that's not what you had in mind