If he's extremely bright, I really would want him to go a to a university where his capabilities are challenged... Would it be impossible to get into Cambridge? (Sorry I have no idea...) That would have been a University that I could have possibly found been interesting (I just chose the obvious and nearest one).
A question out of curiosity: does the international school not offer the possibility of preparing for the Swiss Federal Matura? Or did the student not think about taking it? It's a shame if they don't; I think that any school in Switzerland ought to offer such a possibility, so that a student can continue his studies in the country without too much hassle...
Is it really that impossible? Above you mentioned that the student is fluent in German (apart from technical terms)... why not try? And who knows, the examiners might actually allow the student to use English technical terms, as long as he can talk/write the general part in German. After all, the student would be tested on his knowledge of the subject, not on his German skills in it.
One of my housemates when I was a PhD student in the UK, was an Italian guy who had studied at an international school in Italy, and then went to the UK for university at the age of 18 (his younger brother did the same thing a few years later). I agree that it's not for everybody, but if the student is up to it, why not try, if money allows it.
For UZH vs. Imperial/UCL/Warwick, I might be -maybe- tempted to consider the latter universities of higher quality. Unfortunately it's an impression that is not based on any fact. Although it might be worth pointing out that Imperial is part of the IDEA League ( http://www.idealeague.org/ ) together with ETHZ, so maybe that's worth something.
They don't have to pass it. All they need is a cantonal or federal Matura in which their grade in German is above a certain number. I can't remember what the minimum grade is.
In any case, I think that the requirement for a C1 level is above all a way to ensure that there aren't foreign students with insufficient German skills to cope with the study. And for sure in Maths it's not paramount to possess excellent German skills. (It would be much more important in, say, Law). Even if the student didn't have a C1 level (but a reasonably good one to cope with the study) I would consider contacting the people at the university to explain the situation (student's family lives in Switzerland) and to ask them if it would be possible to have an interview and decide on the basis of the case.
In my first year as a PhD student, I did some teaching assistant work for some foundational courses in Maths at a non-Oxbridge top-something British university which also ranked within the world's top 100, according to these rankings. I was so disappointed by the low level of most students, but even more by the fact that teachers weren't expecting a higher level of them (this might have had something to do with the relationship between University ranking and government funding), that I stopped doing TA work. So, while it is definitely possible to become a *really* good mathematician/graduate even at a university that is in the lower part of the top 100, I would prefer very bright students to go to universities within the, say, top 50. (Having said that, I don't have a very good opinion of these rankings.)
This is exactly the kind of information I don't have, and the reason why I pointed out that my experience was a long time ago. Do investigate with people (such as Laertes) who have a experience with how things work now. Good luck!