Hamin,
No one cares at all by the time you're about 25 or 28 where on earth you went to university in most cases, especially if you've got a good degree. And if you want to go into academia, you find a lot of the dons did their first degrees elsewhere. In some subjects, it's a bit like Harvard, go there to do your masters.
And I regret to say that some employers regard Oxbridge as for toffs, so you can have to live it down.
Also, if you get a good course and good teaching from a good university, you may do better than if you find yourself on the wrong course with an unsympathetic tutor in Oxbridge as has recently happened to the son of a friend.
Finally, and I don't want to rain on anyone's parade, but after the very very obvious top ten percent or fewer of candidates, it's so v heavily competed that it's hard to draw a line: it's not a lottery but there are truly huge numbers of excellent candidates out there, which just means that other top RG HEIs get them.
In fact, for the subject DS is hoping to read, quite a few of the academics in that subject think a particular non Oxbridge RG is better than Oxbridge.
Hope that helps. P M me if you want.