I wrote a fairly long reply to this but accidentally deleted it, so the long and short of it is: yes, the Oxford CS course (including the joint school in philosophy) is fairly mathematical. Even among the more "applied" options they typically offer, they skew on the mathematical side of CS (e.g. graphics, AI, machine learning, databases), and a lot of the options otherwise are in theoretical CS areas (e.g. complexity, quantum information, things like lambda calculus, logic, and category theory). Several of the CS papers are available as options normally to maths degree students there (including those not on the joint maths & CS course) and a few maths papers (that would normally be taken by maths degree students) are available to CS students (both on and outside of the joint course noted above), so they anticipate clearly a fairly high level of mathematical ability in their students.
More generally, any CS degree will include at least some mathematical content, and realistically no degree that calls itself computer science will be functionally a degree in programming. Programming/coding is just one part of a fairly broad subject area that makes up CS, and while an indispensable method for computer scientists isn't necessarily an end in of itself. If you are primarily interested in the coding aspect of what you've done in CS so far, then you might want to look for courses more focused on that (e.g. IT(MB), games programming, applied computing, software development, etc) or degree apprenticeships in the computing sector. It is worth noting though that especially in those CS courses that are generally considered as "top ranking" (e.g. Oxbridge/Edinburgh/Imperial), they will be fairly mathematical and more focused on theoretical topics following from the academic study/research of the field.
Also on the philosophy side, CSP students at Oxford take an extended version of the first year logic course (along with maths and philosophy and PhysPhil students), so even in that side of the course it will be a little more mathematical than the average philosophy course. After first year though you can not take any more logic if you so choose, although I get the impression the logic options are quite commonly chosen by CSP (and maths & phil/PhysPhil) students due to overlapping/cumulative content with the topics in the other subject. There are actually several logic focused options offered by the CS department itself, so you could potentially do a very logic/foundations oriented course in CSP!