I know many Oxbridge and non-Oxbridge PhD students, and Oxbridge and non-Oxbridge postdocs, including people who moved between Oxbridge and elsewhere within their PhD and after their PhD for their first postdoc. They all have a fairly standard educational background.
You also seem ignorant of the fact that you do not apply to a university to do a PhD like it is an undergraduate or postgraduate taught course. You apply to work with a specific supervisor on a specific project, if a university does not have a PI who is looking for a PhD student in the field you are interested in there is nothing for you to apply for there. There are many people who wouldn't even try to apply for a PhD at Oxbridge because none of their researchers are relevant to their own research interests.
You either have no experience in academia or need to get a new supervisor because they are advising you horribly for when you start looking for a postdoc. A PhD student from the bottom-ranked university with a strong publication record, relevant skills, and a well-respected supervisor who gets on with the prospective PI (though by far the first two matter the most - and many would argue they are all that matter) will get a postdoc over an average Oxbridge PhD student 10 times out of 10.
edit: I should clarify I'm talking about science here, depending on subject humanities PhD students often just publish a book.
"It doesn’t matter whether you got your PhD at glittering Harvard University or a humble regional institution like the University of Ballarat. The supposed prestige of the academic institution has almost no bearing on your long-term success, once other key variables are accounted for.
...
Finally, by far the best predictor of long-term publication success is your early publication record - in other words, the number of papers you’ve published by the time you receive your PhD."
https://theconversation.com/predicting-who-will-publish-or-perish-as-career-academics-18473"Finally, we found a surprisingly weak role for university prestige once the effects of other predictors were accounted for statistically (figure 2). This finding held even when we used two other leading university-ranking systems (see the supplemental material), which indicates that it was not merely an artifact of the ranking system that we employed (i.e., the Academic Ranking of World Universities). One possibility is that individual mentors or lab environments vary widely and are more important than is university reputation in determining long-term publication success. Alternatively, key personal attributes, such as motivation, might vary so widely among individuals that they simply swamp the effects of university ranking. Whatever the expla- nation, our findings suggest that, if two job candidates in biology have comparable publication records, there would be little justification for automatically favoring the candidate from the more-prestigious university."
http://www.jstor.org/discover/10.1525/bio.2013.63.10.9?uid=3738032&uid=2&uid=4&sid=21103168668827You sound like an undergrad.