For example: Tuition fees are in line with government-set rates and are no different for these institutions than other British universities (at least it was that way, back when I was there in 2005-2009). While the U.K. was part of the E.U. these low rates even applied to students coming to study there from elsewhere in the E.U. It is only "overseas" students who are asked to pay significantly more.
One thing that I found noteworthy when applying was also the fact that, for Cambridge, the admissions process had two phases: The first was academic acceptance. Those application forms made absolutely no mention of money. After a student has been academically accepted, there is a second phase in the application process that involves application to scholarships and ends with the applicant having to provide proof that they have the required funds to sustain themselves through the study period.
This was very different from what I remember about applying to universities in the U.S. which asked you to disclose your family's financial situation on the same application form that was also used to decide academic acceptance. That form, incidentally, also asked about things like family members who have attended the same university, etc. ...so there is much less transparency there to guarantee that a U.S. school wouldn't just use your socioeconomic standing directly as a criterion to decide admission.
There are, of course, impediments to social mobility into the class of Oxbridge-educated people, but it revolves more around academics than about money. For example, I found that the students disproportionately came from families where their parents were career academics. The relationship with those families having a lot of money was not something I particularly noticed.
The "poshness" of it all, particularly where undergraduate circles are concerned, also comes to a large extent from free choices that students themselves make to try to socially set themselves apart from their peers at other unversities. For example many willfully adopt "received pronunciation" to replace their natural dialects.