Maggie, just to give another angle- I grew up in Sweden which is also popularly supposed to be a classless society. Well, it isn't: some people have more money than others, some have nicer houses, some are more likely to go to the opera and encourage their children to aim at higher education. There are very clear differences between nice leafy suburbs and less desirable areas and this feeling is growing.
But at the same time, there are big differences from the UK:
There is no assumption that if you belong to a certain class, you will automatically be less accomplished in the aspects of life that require simple common sense, such as child rearing. I have never had the same sense there that we of the middle class need to teach the lower classes how to behave around their children.
There is no assumption that if you belong to a certain class, you will be less able to do practical things such as cook a nutritious meal.
You may, of course, have less Latin, or less familiarity with higher education, but there is no sense that this will influence your general ability to deal with life.
And it is assumed that all parents (unless they are positively anti-social) have valuable practical skills to confer on their children.
Pretty well everybody attends the same schools, so there is very little in the way of an upper class which have a totally different way of speech and set of experiences. (the new independent schools are generally speaking not elite schools)
Many pursuits popular in the country- such as swimming (enough lakes and beaches for everyone), forest walks (right of public access), berry picking, skiing, skating, fishing and hunting- have no class connotations and don't cost very much, which means there can be common ground between people of very different backgrounds.