I think sadly that social mobility has fallen but I don't think that's anything this govt has done.
I think there has been a huge democratisation of success - ordinary families now expect to send their children to university and buy their own homes. This is great and is a far cry from the society my father grew up in (working class, most kids left school at 15, no university aspirations, never dreamed of owning a home. In those days getting a council house was a major aspiration - my grandma begged for one to get her family out of the desperate hovel they lived in).
The flip side of that is that with a majority of people owning homes, prices are bid up because there is now a market for houses (whereas before a house was just something you lived in).
And with so many children going to university, those who don't go are left far behind the majority (whereas previously most didn't go and so you weren't behind the majority of society if you didn't go).
These are vast social changes that are difficult for any government to manage. I don't see that Labour has made things any worse - its policies are aimed at removing any remaining barriers where it finds them hence combatting child poverty, Surestart etc (though I would prefer that universities remained free ).