Tl;tr: basically, yes, there's a kind of caste system, and it's hard to escape where you come from. Mobility has replaced nobility to an extent, but people are generally happy with, and proud of, being working class, and there's a bond between working class people that's stronger than geographic bonds. If anything, people are often guilty of their privilege, and try to play it down.

I'd say the comments about the Travelling community ring true. They are vilified in the press, and communities will rise up in opposition if a Travellers' site is proposed nearby. But they also have their own traditions and ways of life, and don't particularly envy the house-dwellers. You do hear stories of Travellers "settling down" and just hating it.

The upper classes (aristocracy) are largely the subject of ridicule among the vast majority of Brits. The world has moved on without them, and the things that once made them rich (basically owning land and its people) no longer apply. Most of those that survive are either poor or have sold their property to the National Trust. There are exceptions (e.g. the Duke of Westminster) but they're rare. The billionaire class in the UK is mainly new money now - people like Dyson, Ratcliffe and Branson.

In between, you've got the middle and working class. "Middle class" doesn't mean the same as it does in the USA. It's working people, but they're in the "respectable" trades - law, medicine, finance, local and national government etc. They might use state education rather than private, but their wealth allows them to move into the catchment areas of the best schools (it's a factor in house prices) and/or have their kids privately tutored to ease them into grammar schools with entrance exams.

The working classes are traditionally manual labourers and those in the service industries. They will almost invariably have a state education, with a choice of schools that's largely restricted by their location, which will tend to be in the "poorer" parts of town.

But there is definitely something else, which is hard to put your finger on. The young middle class have opportunities that working classes don't, through education, contacts and ability to take financial risks. Working class kids who become rich adults are more likely to have done it through hard work and talent than just taking a load of risk-free punts and investments. It can take several generations for the accents to disappear though. People who know know - and it can be both negative and positive, in both directions.

There's a weird phenomenon in Britain where middle class people downplay their comfortable upbringings, and pretend to be working class. You'll see dozens of examples in any university bar. There's a kind of "purity test", especially in the music industry, where working class artists are taken more seriously, and it's quite embarrassing when it turns out that the latest indie hopeful has a trust fund or a private education. A lot of journalists, especially those representing the Left, also like to be seen as lower in class than they actually are.

I suppose this all adds up to a caste system of sorts, but it's soft-edged, and the people within the "castes" generally don't particularly want to move out of it. Everyone would like more money, but it won't change your class. However, it's far, far away from the places where actual caste systems pertain.

Fuck me, that went on a bit.