I lived in London for four years. Perhaps I just didn't appreciate it, but I can't say I really was that aware of the class system on a day-to-day basis.
Sure, you were aware that in Peckham in South London there tended to be a lower socio-economic 'category' of people than say, Kensington. And yes in Mayfair there were the private members' clubs and the like. And certainly the race day events tended to attract certain people.
However, I, and others like myself who were there for a few years (from NZ, Aus, South Africa, Canada and more) didn't see it as a limiting factor. If we wanted to go to the races, we would, and just dress up for the occasion. Bond Street is just a place to shop, the prices are higher, is all. Nothing stopping you strolling along and having a look.
It's funny, every so often a comment would crop up about it. Someone at work would make a comment about not taking the bus because they weren't poor enough to need it (or the tube). Yet, the current Mayor of London bikes to work, and the previous Mayor used to take the tube (still does, last I checked). And most of us Kiwis or Aussies etc, whether from rich or poor backgrounds - we'll take buses, tubes without hesitation - they're quicker, convenient, and driving in London is awful.
'Class' certainly didn't stop me getting into the Buckingham Palace garden tea party :) In general, the UK has become (well, London, but other cities more and more as well) so cosmopolitan and multicultural that in some facets of society, the class system has all but melted away. That's not to say that some people don't still hold it true and fast and insist on it and campaign against it when a fast food restaurant wants to open in Highbury (it'll lower their class somehow?), but on the whole, most people you meet are welcoming no matter your background.
As some people have said, in London - nobody's a foreigner, because everyone's a foreigner!