Cambridge does explicitly state that it is not possible to take Maths jointly with any other course. However, I was not aware that there is the option to take Computer Science with Maths in the first year. I looked it up and this seems to be not an independent course but rather an option for the Computer Science course. Sorry for the misunderstanding.
Apparently you have to apply specifically for CompSci with Maths if you want to take this option in the first year. From the course description all Computer Science students seem to have the same subjects and options in the remaining years of the course, please correct me if I’m wrong. This would mean that you would study mainly Computer Science apart from first year. On the other hand, at Oxford Maths + Computer Science is a seperate course and it is half CompSci half Maths for the whole duration. How much Maths do you want to study? That’s probably one of the most important questions you have to ask yourself when making the decision. As I said before, take a careful look at the course descriptions. Also, if I have misunderstood the Cambridge course setup somehow, I’m sorry, I’m more familiar with the Oxford courses and Maths in general.
Another thing you should be aware of are the different application processes. For Oxford you would have to take MAT, the Maths admission test (but you get to do a Logic question inteded for Computer Science students), and from what I have read on Cambridge’s website you will be asked to do STEP if you decide to do Computer Science with Maths. STEP II + STEP III are difficults Maths exams which are taken by all Maths offer holders and some NatSci, Engineering and CompSci offer holders. However, if you decide to study pure CompSci you would not usually have to do STEP, while MAT would still be compulsory at Oxford.
I said that the amount of Maths you want to have covered should be crucial in making your decision. However, be aware that your capability of handling STEP should be a deciding factor as well, as MAT and STEP are rather different exams.
MAT is taken in November and used to shortlist for interviews. At that point in time the pool of applicants is still large and MAT is used to narrow down the number of applicant to three per place. It has a multiple choice section and four long questions. If you are applying for Maths + CompSci you will do three of the Maths questions and one of the CompSci questions. Have a look at past papers, they can be found on the Maths Institute website. After that your interview counts.
Cambridge does things the other way around. First almost all applicants are invited to interviews and then offers are given. However, half of the Maths offer holders fail to meet the requirements due to STEP, I don’t know what the situation for CompSci with Maths is like. STEP is much harder than MAT, because MAT is taken by all applicants prior to interviews while STEP is taken months later and only by offer holders. STEP is also not adjusted for CompSci students, it’s purely an exam made for Maths applicants. If you decide to apply to Cambridge have a look at the past STEP papers to see if you can handle them. Applicants typically also practice for months before taking STEP. (You should, of course, also practice for MAT if you apply to Oxford.)
Good luck!