Its not hard to find out. Look at the official statistics published, not what people on here 'perceive' to be the poshest.
Every year the Sunday Times ranking has a list of the unis with the lowest state school intake, and also the most middle class. Also HEFCE (the gov body) every year produces data on which unis have the lowest % state school intake. The data should be on their website. Also, the websites of the top public schools like Harrow, St Pauls, Radley etc list where their students go to university every year.
From memory, the same suspects are usually in the top 10 in the same sort of order. They are
Oxford
Cambridge
Durham
Bristol
UCL/LSE/Imperial
St Andrews
Edinburgh
Nottingham
Bath.
Not sure why everyone thinks Exeter is so 'posh'. It isn't usually in the top 10. After these few then the next 'usual suspects' in recent years have tended to be Exeter, Newcastle, and Leeds. A lot of it is about fashionability. For example, Nottingham is loved by a lot of big public schools especially the ones in the green belt, and they send a huge number there every year (see university destinations pages on public school websites). Alternatively, similar universities rep-wise like York and Warwick dont even get a look in.
If you want to avoid a rah uni, then go to a big civic like Manchester, B'ham, Sheffield.