The way the world sees things - rightly or wrongly - places Oxbridge at the pinnacle of the British university hierarchy. Successive rankings of British universities over many years have validated this view - although it may be that the ranking methodology has an in-built bias which tends to produce this result, so I wouldn't set too much store by that. The reality, though, is that even if a purely objective, unbiased ranking were possible, and showed that other British universities were equalling or surpassing the Oxbridge universities, it would take a very long time for public opinion to change in the matter. Public opinion generally trails well behind the evidence.
While this remains, the competition to get into Oxbridge will always be stiffer than it will be to get into other universities ... and those who have succeeded in gaining entrance (as well, it has to be said, as those who have not) will be keenly aware of the fact that they have been successful in a competition to demonstrate that they have those qualities which Oxbridge colleges look for in the students they select, whereas others have not.
Whether this means that they "look down" on those others who competed unsuccessfully, or did not compete at all, is another matter altogether. That is down to the individual personalities of the students themselves and, inevitably, there will be some who will and others who will not. It is not something about which you can generalise.
However, there IS an old joke which goes like this:
Q: How do you know if somebody went to Oxbridge?
A: They tell you
This joke doubtless reflects reality to some extent. It is tough getting into Oxbridge, not all those who try it succeed, and those who do succeed tend to feel a (justifiable) pride in having done so. Pride need not involve condescension and looking down on others; but sometimes, inevitably, it does.