There is a lot of "it depends".
I agree with Ianica58 that engagement and wider interests will get you a long way for many careers.
Yes private schools love plenty of Oxon or Cantab on their staff lists, and Magic Circle law firms may use Oxbridge as a recruitment shortcut. Senior Americans often put a lot of emphasis on the "school" you went to, but will normally put the better known London Universities on the same footing.
For the last couple of years I have let to students at Bournemouth University, and both groups have been super. If I was an employer and skill sets matched, I would have no hesitation recruiting them. Ditto the BU marketing graduate working for the agency who looks at a room, and explains what needs to be added (pictures, cushions etc) before she comes back to take super photos for their attractive website. Or the BU law graduate who specialised in age restricted leases, matching his ability to support and protect vulnerable clients.
Once in, I don't think it makes much difference. Its about merit. DH had a senior City colleague who climbed the ladder starting with BU and then up via local backroom operations and then a transfer to London. One of his colleagues did similar starting at a new University in Yorkshire.
But again it depends. A naturally academic student may love a arts/humanities degree at Oxbridge and this degree may then open doors to compatible and rewarding careers, whether well paid or not. DD (who is dyslexic) decided very early on that she did not want to work in an office. She had a bit of an academic breakthrough aged 15 which meant she is studying medicine rather than pursue original ideas of physiotherapy or PE teaching. She, rightly, decided not to apply to Oxbridge, even though she would have been a credible applicant, heading instead for a more hands on course.
DS is happy in a library. My plumber, in his mid 20s, skipped University, in favour of a vocational degree. He probably earns £200,000 pa. They met last week as DS was heading off for a Skype interview for a research assistant position at a well known Ivy. Plenty of mutual respect but DS declined the plumber's offer to teach him how to change a washer. DD in contrast can change washers, unblock toilets, bake bread, and can apply her first aid training. She would have been quizzing the plumber about gas pressures, salary etc.
At the end of the day, Oxbridge may help open some doors, but will not entitle you to anything. It is worth building other skills in parallel.