Okay then, Swedes, in summary I would suggest that changing schools may or may not be the right thing to do, depending on your child's feelings and the profiles of the schools concerned. What I would not do, is to assume that one school must be 'better' because it has a much higher success rate, since this could easily be related to the profile of the pupils or the ambitions of the parents.
What I would do is to look closely at the profiles of all the possible schools for the subjects your ds is interested in. Eg, how many subject specialists do they have, is any A-level teaching done by non-specialists? How many students take each subject and what are the grades like? What subject-related extra-curricular activities do they do? Etc. Go to open days, talk to staff and students and look at the facilities. If one school is clearly inspirational in the area that your ds is going to be doing, then it makes sense to think about moving, if he likes the idea.
Meanwhile, if he's serious about wanting to do Oxbridge in a particular subject, he needs to immerse himself in that topic way beyond the requirements of the curriculum. For example, my dd1 (12) is convinced she wants to read Vet Science at Cambridge, or failing that at Edinburgh(!). She may well end up doing something different, but if she is still interested at 14, I would be encouraging her to think about shadowing a couple of vets, and later doing voluntary work in one of the animal clinic charities, skivvying for a practice nurse, or helping out at the zoo or on a farm, all in addition to reading widely on the subject.
Prospective medical students absolutely need to have done voluntary work in a range of settings to convince the selection panel that they understand what is involved and have the commitment or the stamina to see it through.
English students need to have read a lot of books, seen lots of plays and films, and have serious, informed opinions on them.
In the end, this is the kind of preparation that will get him where he wants to be, assuming he has the ability. Assuming all the schools you're choosing between are good, the choice of school is only ever going to be a secondary factor.
hth