The fact that you had a Cambridge offer suggest you are a worthy Cambridge candidate.
Here is my advice. DO NOT GIVE UP.
Some young people give in on their dreams, and go to a uni they didn't want to, or study a course they didn't want to, because they did not have the strength to take a year out, and try again and resit. That ONE tiny year extra, seems to make so much of a difference to some people, and I honestly have no idea why. Sacrificing going to Cambridge for the sake of wanting to go to university the same time as everyone else is incredibly foolish. A year longer in the long run has absolutely no effect on your life, but going to a university which you did not want to go to, can have a substantial effect.
So here is my advice. Do you want to study at Cambridge, or were you pushed into it? Do you want to go to that local uni or do you feel pressured into going to university straight away, the same time as everyone else? Do you feel that you could have performed better, but your mental struggles pushed you back?
Because if you genuinely want to go to Cambridge, and think you are deserving (which I am sure you are, you had an offer) then DO NOT GIVE UP. Take a year out, resit, go to another college and start afresh if you want, and then reapply. Try and seek help for your hypochondria and anxiety, because if you are having U's now due to your mental health problems, are you ready to go to uni? Help is there for you, so many good people on here can advise you on that.
But equally, if you are happy with going to the local uni, and were forced into applying for Cambridge, then very kindly stick two fingers up to your family, pack your bags, and enjoy yourself. But I'd also still recommend seeking some help for your mental health problems.
It's your ****ing life. Do what YOU WANT. Don't let your family, and don't let a measly year, effect your entire life.