I don't think that's actually a fact. I don't know what figures you're using, but eyeballing the Global Social Mobility Index puts the UK as pretty middling in Europe. Less mobile than France or Germany, more so than Italy, Spain, or Poland.
It's below average, and lower than all of Scandinavia, Austria etc. Yes, it beats Southern Europe and ex-communist countries.
My source was the Sutton Trust and Oxera.
well, that's not one of the lowest really is it
"One of"
If your point is the UK is just about in the lowest 50% of European countries by social mobility (keeping in mind Europe is pretty good on a global scale for social mobility), it's basically a non-point altogether. If the UK were at the end of the scale it might, but it's not, so it doesn't.
I think that if you look at the people in positions of control in political and economic sectors (and mass media, which is a key element), the vast majority came from just a few schools.