Wow, didn't think I would get responses from people who are actually from the UK. Thanks so much guys!
Sounds like doing an MPhil would be a good place to start for me. It should be easier to get in, and I hope it will provide directions for my PhD research. I'll certainly apply to Cambridge and Oxford, and maybe 3-4 other places like Edinburgh, Imperial, and Sheffield. But honestly, given my undergraduate school, I'd have to hope for the first two. Do you guys think that if I get into, say, the Cambridge MPhil program, applying to its own PhD should be easier?
Also, it would be useful to know what the completion rate for people who start their PhD at Cambridge or Oxford is like. Are there significant hurdles, like qualification exams, too-heavy workload, or lack of research progress, which prevent people from getting their PhD?