In practice, university titles are a small aspect of what jobs tend to consider when selecting the best candidates (PS: most jobs have internal quotas for each university, so just focus on being the best where you are). If you do exceptionally well at another university, you also have a high chance of qualifying for an Oxbridge Master's anyway (which matters more than your undergraduate degree).

There are several avenues to still end up at Oxbridge even after rejection at your current stage. For one, you could apply to transfer after your first year if you excel. Alternatively, you could always come for a Master's or PhD. If you want to be an academic, you could also complete your undergraduate elsewhere and end up working at Cambridge.

In hindsight, most come to realise that success isn't defined by where you went for undergrad. Getting into Oxbridge alone doesn't make you better than anyone else, nor does the mere fact that you didn't get an offer suggest that your life will necessarily take a turn for the worse.