I’m at New College (and study philosophy as part of PPE). There’s some truth to what elilast said about the cliquey private school element, though it’s certainly not overbearing or inescapable (so long as you don’t play rugby lol). In general, my experience has been very positive. I would say it’s a very fun, outgoing college but also very academic — definitely a work hard, play hard kind of vibe. One of the college’s biggest pluses (aside from it being absolutely gorgeous) is the accommodation, which is arguably of the overall highest standard of any college. The clear majority of rooms are en suite, and they’ve just built brand new third year accommodation, which means that you can now live in college for your entire degree.
New College is also a really strong college for philosophy, with great tutors (three philosophy fellows, which I reckon must be about the highest of any college). One of them is Volker Halbach, who’s the leading logician in the Oxford philosophy department, which might be of particular interest to a CS and phil student. My interests are more in moral philosophy, so I’ve been taught more by Stephen Mulhall, who’s also fantastic. The only thing to note in your specific case is that the college only admits CS and phil students sporadically, so it’s quite likely if you got in you’d be the only one — though there’d still of course be other computer science single honour students, and loads of others doing the other philosophy joint schools.