Oxford is not the be all and end all, especially not in medicine. If you don’t think it’s going to be the right evironment for you, don’t go - you will get the same GMC-accredited degree from any other medical in the UK. Your mental health is more important and from what I know about Oxford, it is not the most conducive environment to good mental health, and spending 6 years in such an environment when you are already struggling is likely to be very tough.
I never considered Oxbridge for this reason even though my school encourage me to apply, and I have no regrets - went to a much chiller med school, had a decent time and graduated with prizes, presentations at conferences and all the other CV fodder that I needed further down the line for specialty applications after the foundation programme. Still struggled with mental health, but things were manageable - I don’t think they would have been if I’d gone to Oxbridge.
I think if I was in your shoes, I would probably turn down the Oxford offer in favour of Brimingham, or potentially defer the Oxford offer. If you are dead set on medicine, I think it would be very foolish to turn down both offers with a plan to reapply for a London uni because there is no guarantee you will get a place (you are clearly very able and a good candidate but with medicine, there are never any guarantees - having one bad day on your interview is enough to cost you an offer, even if you were the perfect candidate up to that point).
Whatever you decidie to do, best of luck, and I hope that you are getting help with your mental health and have supportive people around you to help you through this.