To be honest, I think the majority of people who like the OP are perfectionists and get anxious about their work will be the same at any other top university. Even though other universities normally have fewer deadlines, you'll still be putting pressure on yourself throughout the course if that's the way you are.
I've never been a perfectionist, had any mental problems, nor been a naturally anxious person when it comes to my work, and I don't think Cambridge has changed this. I'm still pretty laidback and rarely get stressed out about work. The work is harder, definitely, and I myself work harder, but the only time I can recall being really stressed out about the work is in the few days before my exams last year started.
Maybe other people disagree, and think Oxbridge causes/worsens mental health problems, but I think university in general could do this, rather than specifically Oxbridge. The OP mentions being a perfectionist, and as others have mentioned, your best is never good enough. Your best might get you a good grade, but there is always something to be improved upon. Like Jcb, I find the lack of specific tasks leading up to an essay quite challenging (I do philosophy). At A-level, I just had to read the textbook/set text and then write the essay. Here, we have a long reading list which covers a broad topic, but the essay questions set are so specific that often I'll read a lot of the reading list but barely anything is relevant. When it comes to revision, I don't have time to read the whole reading list, so I start to worry that something will come up in the exam for which I have done no relevant reading. So I'd say if you're an arts student that might be something to consider. Sometimes I feel if I'd gone elsewhere, I might have been (academically) happier as a lot of philosophy exams at other universities are based heavily on lectures, with specific reading lists for each lecture.