Do accents still play a role in British class distinctions to the present day? How have things changed since the 1960s and Received Pronunciation?

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?

Things have certainly changed in that, for example, you find more TV presenters with regional English accents in more "serious" roles on national UK television. On the other hand, national news programmes still tend to be fronted by presenters with what are perceived as essentially "standard" English accents. Perhaps tellingly, it's been for some time common for national presenters to have Scottish, Welsh or Irish accents, but not regional English accents.

Figures in the public eye with notable regional accents such as John Prescott and William Hague have visibly attempted (with varying degrees of success and ridicule) to "iron out" (i.e. move more towards something like RP) their accent when speaking publicly. It's not clear to what degree this is conscious or subconscious, but either way, it's telling of our perception to accents that they do so.

Note that what is perceived as a "standard, non-regional" accent of English is almost certainly no longer RP as traditionally transcribed in EFL textbooks (assuming the principle of transcribing vowel sounds with the nearest cardinal vowel symbol). For example, the fronting of the /u/ vowel (so that it sounds closer to French "i" or "u" vowels)-- a phenomenon that is probably at least a century old-- now seems to be fairly standard, but is practically never reflected in general transcriptions in dictionaries, EFL textbooks etc.

It's worth considering that the British notion of "class" has probably changed somewhat in the last few decades as well. We live in a world where athletes are given knighthoods and Floella Benjamin is a baroness.

Kate Fox is an anthropologist rather than a linguist, so I'm dubious of some of her descriptions of accentual differences, but it's easier to distinguish accents than to explain them so there's still some value in quoting:

There is, however, a distinction between upper-class speech and 'educated' speech -- they are not necessarily the same thing. What you may hear referred to as 'BBC English' or 'Oxford English' [i.e. R.P.] is a kind of 'educated' speech -- but it is more upper-middle than upper: it lacks the haw-haw tones, vowel swallowing and pronoun-phobia of upper-class speech, and is certainly more intelligible to the uninitiated.

Watching the English, Kate Fox, pp74f

And to pick up our discussion on class from the comments, because this wouldn't fit there, from p82 in the summary of the chapter on language:

The linguistic codes we have identified indicate that class in England has nothing to do with money, and very little to do with occupation. Speech is all-important. A person with an upper-class accent, using upper-class terminology, will be recognised as upper-class even if he or she is earning poverty-line wages, doing grubby menial work and living in a run-down council flat. Or even unemployed, destitute and homeless. Equally a person with working-class pronunciation, who calls his sofa a settee, and his midday meal 'dinner', will be identified as working class even if he is a multi-millionaire living in a grand country house. There are other class indicators -- such as one's taste in clothes, furniture, decoration, cars, pets, books, hobbies, food and drink -- but speech is the most immediate and the most obvious.
...
This reliance on linguistic signals, and the irrelevance of wealth and occupation as class indicators, also reminds us that our culture is not a meritocracy. Your accent and terminology reveal the class you were born into and raised in, not anything you have achieved through your own talents or efforts.

I would say that accents and speech certainly do still play a role in class distinctions, assuming (and this is a big assumption) that class distinctions are still relevant in contemporary UK society. In general, the lines between class and accent have become blurred somewhat.

I don't think we actually need to discuss the role of RP in its traditional, 1950s BBC English sense, since it is so rarely heard and verges on anachronistic today. The standard 'English' accent is a more toned-down version of this – something closer to a refined Estuary English from the South-East of England around London. Many UK politicians from very educated, perhaps what might be described as upper-class backgrounds, have adopted this Estuary English accent in order to hide their background, just as presenters or politicians with very strong regional English accents have perhaps modified them towards this middle ground of "refined Estuary English". It appears that British accents are converging on a middle ground. Even the Queen is said to have 'toned-down' her RP-accent since she first started making television/radio addresses in the 1950s/60s.

While you do hear regional English accents on TV nowadays, you very rarely hear broad, dialect-heavy speech. It is more often than not a very toned down version of whatever regional/national variety of English it happens to be, and this includes Scottish and Welsh presenters and politicians.

The marker of class then (the establishment class?), I would say is conformity towards some sort of middle ground around a British 'standard English' based on Estuary English. A regional accent by itself is not enough to distinguish class, though it may once have been. Other characteristics of speech such as vocabulary, certain colloquial usages etc. etc. may also be subtle markers of background or education level. (I must point out that I dislike using the word class as it's becoming less relevant each generation)