Have you checked that the course you want is available part-time or on a flexible basis?
I don't know if it's different now, or if it varies by subject area, but I did an MPhil at Oxbridge about seven years ago and they were still really archaic about part-time or flexible study. It just didn't happen. You were supposed to be full-time, fully committed (no working on the side) and prove you had the funds up-front.
After my MPhil, I was accepted for a PhD there too, but at the last minute had to drop out, as I didn't get research council funding and there was no option to self-fund as you went along: you had to show a lump sum, or grant funding, to start the course. I believe someone more enterprising than me actually sued them about this a few years ago, and won, so maybe they've adjusted their policies and become a bit more flexible, both about finance and courses in general. I would be surprised, but here's hoping.
There are part-time courses at Oxford and Cambridge, but not in all subjects.
I believe you are still asked to show your funding up front, but can argue your case (actually, IME, you could do that before that bloke sued, but they didn't publicize it and it was pretty ad hoc). Most universities do ask you to show how you're going to fund yourself. It's not just Oxbridge.
You might want to contact the Continuing Education department, at least at Oxford. They do a lot of courses, leading to all kinds of qualifications. It is do-able, a lot of students do it every year, but it does depend on you being able to commit the time. (Ruby Wax had to get an extension, to complete hers.)
Why do you want to do it? Will it really increase employability? etc.
The Dept of Education at Oxford is one of the only (non contend) departments at Oxford that offers part time MSc's and DPhil's. I did my (2nd) MSc there whilst working full time. I did Educational Research Methodology,it took two years and was bloody hard work. I didn't find the uni particularly geared towards p/t students, for example the libraries are often shut at a time a p/t student would want to use them, a lot of seminars, talks, networking events are in the evening (not great for family life), and of course you have to get leave for the times classes are scheduled, and that changes on a termly basis.You also have to pay college fees as well as course fees, and student life revolves around the college. But you would probably go to Kellogg which takes P/T students, and is a lot more relaxed than some others. Saying all this though, it certainly has given my cv a boost and was worth every penny.