An Englishman's way of speaking absolutely classifies him.
The moment he talks he makes some other Englishman despise him.
If you spoke as she does, sir,
Instead of the way you do,
Why, you might be selling flowers, too!
Those are probably my two favorite lines in my favorite song of my favorite musical, My Fair Lady. I have to admit I've been caught quoting them once or twice when asked by friends or family why I tend to be exacting about proper English speech. The movie made huge impression on me since I first watched it at age 6; Henry Higgins is kind of a personal hero of mine.
But how true are his words to reality today? Now, I've read all about Received Pronunciation, and hypercorrective hs, and they are indeed interesting topics to discuss, but I get the impression that not even the upper-class adhere to Received Pronunciation anymore, and that much effort has been invested by many British to be more colloquial in speech so as not to seem outwardly too posh or upper-class. I get the impression that many view speech distinctions as something to be publicly denounced or abhorred. That doesn't mean that those distinctions don't exist of course, but Received Pronunciation in particular seems to me a social distinction of a long past age, and besides there are a ton of other English accents to talk about. Discussing RP as the totality of what it is to be said on the subject also seems myopically centered on London to detriment of the rest of the UK.
I think it's also important to consider that the demographics of the classes and thus the linguistic baggage different ethnic groups brought into British speech might have changed the different distinctions. For example, I know that upper-caste Indians have become a prosperous group in the UK; have they in any way changed the markings of the upper class speech? What about the ascendance of Jews escaping from the Holocaust, and a "Yiddish" manner of speaking they might have brought with them? Have wealthy and prominent Russian moguls changed speech patterns? (For example, in this question the question-answerer remarks on the middle class' willingness to use na zdrovyeh as a toast in place of cheers; have other things changed?)
The converse probably holds too: I'd guess that immigrants from Commonwealth Carribean countries and Polish migrant workers have possibly changed distinctions on what it means to have "working-class" speech patterns. What can be broadly said about all this?
TL; DR summary: What examples can you offer of accents or speech differentiating social classes in the present day that doesn't discuss Received Pronunciation?