As has been said, no pupillages are reserved for Oxbridge graduates. A disproportionately high number of Oxbridge candidates do obtain pupillage, but that is not because of the fact that they went to Oxbridge. The reason for the disparity in numbers is that sets recruit the strongest candidates. Oxford and Cambridge recruit the strongest candidates at an earlier stage, and whilst it doesn't automatically follow that someone who goes to Oxbridge is going to be a strong candidate for pupillage, there is a higher probability of that generally because of the fact that they were strong enough to get into Oxford or Cambridge.
However, the mistake that you're making here is that there is causative connection between the university that you go to and your chances of securing pupillage. There is a correlation, but there is not causation. The strongest candidates secure pupillage. More of those candidates come from Oxbridge proportionally than other universities, but there are still a lot of strong candidates at other universities. Indeed, as you know, the majority of candidates who secure pupillage did not go to Oxbridge at all.
The university that you go to, in and of itself, is irrelevant to your prospects of obtaining pupillage. That's why sets are increasingly anonymising university names on pupillage applications. So as much as you feel like you have ruined your chances, in fact the opposite is true. Your prospects of obtaining pupillage haven't changed at all, because they were never based on the university that you attend. They are based on your own achievements, skills, experience and ability. The university that you attend does have some correlation to that, because strong candidates at the end of undergraduate are likely to have been stronger candidates at A-Level, and stronger candidates at A-Level get into better universities, but as I say, that is correlation and not causation. You are still the same person and candidate as you were before you were rejected from Cambridge. Literally nothing has changed in that regard. There is no reason at all why you can't attend Nottingham, LSE, or a host of other good universities and go on to get pupillage. Whether you get pupillage or not, it will not be down to the university that you attended.
So instead of lamenting not going to Cambridge and getting down about it, use this as motivation to show everyone that you can go on to be a barrister without going to Cambridge. I was rejected from Cambridge too and went on to do exactly that. There's no reason why you can't do the same if you apply yourself properly.