The New York Times today had a piece called "The Paradox of the New Elite," which contained the following statement: "As recently as 1988, some 67 percent of British citizens proudly identified themselves as working class. Now only 24 percent do." I found this surprising--things like that usually change slowly. The article linked to an article in the Guardian, which linked to a BritainThinks poll. That was where the 24% came from. There were some interesting things in the poll report, but at the end it said "This study draws on an online survey of 2003 UK adults. Data were demographically weighted to be representative of the UK adult population." The combination of "online survey" and "demographically weighted" usually means that the survey organization made no attempt to get a representative sample--they just relied on people who could be enticed to their website.
I looked at the British Social Attitudes survey, which does things the old-fashioned way (selecting people at random and contacting them). They have the following question "Most people say they belong either to the middle class or the working class. If you had to make a choice, would you call yourself middle class or working class?" Unfortunately, the most recent year was 2006, but then the results were 56% working class and 35% middle class. Earlier data for the same question can be found from the British Election Studies. They didn't ask the question in 1988 (there was no general election in Britain last year), but 55% said they were working class in 1983 and 57% said they were working class in 1987. No change in 20 years.