I think class is still a huge thing in Britain, but it's incredibly subtle and the class distinctions between working/lower-middle/middle-middle/upper-middle are sometimes more significant than the distinction between the aristocracy and regular people. Though the shadow of the 'old boys network' still casts a shadow over some industries. Being working class isn't necessarily considered a bad or undesirable thing - some 'posh' people (especially in 'hip' industries like the media) pretend to be working class for the cool points, and "middle class" can be a derogatory term - if someone described say the decoration of a house as "very middle class" they'd probably be trying to be insulting. Also, one crucial thing which perhaps not all non-Brits are aware of is that there's no correlation between income and class. There are working class millionaires and upper class paupers - and no one would think someone in the latter category to be less upper class because of their poverty. Though it's likely the child of a working class millionaire may become middle class depending on education and other factors - moving between the classes is complex.
Royalty/aristocracy aren't on most people's radar that much. People might glance at a pic of the royal baby and coo, or have a Royal Wedding celebration, but in day to day terms they aren't that relevant. Kate'n'Wills are treated like any other tabloid-staple celebs except they don't do tabloid-worthy things like Instagram bikini pics or get arrested. I think the Middletons are a good example of British attitudes towards class. Carole was looked down on by the aristocracy for having had a middle class job, whereas regular people couldn't care less. Kate was looked down on by regular people for never having a real job, whereas amongst the aristocracy it would be completely normal for a young woman not to pursue a job or career but just do little bits while waiting to get married.
I POD the recommendation of Kate Fox's book, the very best book about class in Britain. Also Jilly Cooper wrote a non-fiction book called 'Class' which is dated now, but pretty much the definitive work about the UK class system in the 1970s. (I have a PDF of it if anyone would like to read it - I'm not sure how easy it is to get hold of a copy now as it was out of print.) The TV sitcom 'Keeping Up Appearances' is all about British attitudes towards class, especially the relationships between Hyacinth (who is the definitive recently arrived middle-middle woman) and some of the upper-middle acquaintances she's always trying to impress. They got it pretty much bang on. Oh, and the John Betjeman poem 'How To Get On In Society' which is a class satire. A really fantastic book about UK history in general is John O'Farrell's 'An Utterly Impartial History of Britain, or 2000 Years of Upper Class Idiots in Charge.'
Incidentally, my father is from a very old English family - our ancestors were Lords of the Manor in the Middle Ages, though his own upbringing was very humble, and my mother's parents were Eastern European immigrants. So I come from a 'mixed-class' background which is faintly interesting. At least, I think it's given me an interesting perspective on class.