certainly the post war grammar school system, and that of assisted places, had quite a profound effect on social mobility.
And changes to the economy. It simply wasn't possible for the pre-war class system to remain unchanged, because even in 1945 it was entirely obvious that there would be a massive increase in the requirement for people with an education. The war had been won in large part by technology and science (Chain Home, Ultra, Oboe, Gee, Tube Alloys/Manhattan Project, etc) and that was going to drive the post-war economy. Governments had gone from casual laissez-faire amateurism to a massively interventionist civil service in which "the man in Whitehall" was quite keen to know better.
The expansion of the NHS, post-14 education, enlargement of government, the cold war: all of it needed more people with educations to do white collar jobs than could be provided by the pre-war educational system, hidebound by class.
One interpretation of what happened after 1945 was that the country opened up to social mobility because of a progressive, forward-looking desire to equalise opportunity and open up the historic professions to the whole of society. You have to believe in Father Christmas to take that seriously.
Alternatively, one might argue that the elite retained all the power they had before, and provided just enough "opportunity" to get the resources they needed (an educated middle-class) whom they exploited just as shamelessly as they had exploited the resources they needed (a huge manual labouring force) before the war.
And by setting up conflict and jealousy between the historic working classes and the new middle class, the elite could divide and rule the wage-earning class ("workers with hand and brain"). The new middle class saw themselves as above the "working class", and therefore wouldn't worry about miners and rail workers being shafted, while the miners and rail workers returned the complement. In reality, neither had any industrial power, and degree'd school teachers and local authority planners were precisely as vulnerable to changes in government policy as the people down t'pit and in t'shed. The elite always win.