Big hugs to funnyperson...
I am absolutely certain that Oxford ( or indeed any other university) wouldn't be against the child of a broken home. If anything it's another plus in their eyes for your DD - she's got herself to interview stage ( great exam results, extra curricular reading etc etc) despite personal difficulties and upheaval. That is no mean achievement. If there had been particular problems for her during the last few years might her school have mentioned them in the reference?
Personally I'm not sure how or what you would tell Oxford about lack of support at your ex's home - your DD has obviously managed ( excelled even) despite no desk, no stationary and no encouragement from one of her homes. I know you want the admissions tutors to appreciate that your daughter is not one of the spoonfed, privileged ones and that she has arguably worked hardder than some to get this far, but I honestly think they will get that through talking to her. Isn't that the purpose of the interview after all? Was it Incognito Interrogator who posted that s/he didn't want to see applicants dressed as if by their parents nor coming across as over prepared by school and woefully underprepared by themselves.
At 2 a.m when you posted everything seems a little bleak. This Oxford lark is a little bit of a lottery - I'm sure almost every single applicant who makes it to interview ( and a good few more besides who "fail" at an earlier hurdle) would do well at Oxford/Cambridge, assuming a genuine interest for their subject, but sadly not all can be accommodated. I'm telling myself and my DS that we have to roll with the decisions of the admissions tutors and it's no "fault" of his if something didn't click with a particular interviewer on the particular day he was interviewed. He LOVES politics and economics, has become really interested and knowledgeable about philosophy all through his own reading and efforts but who knows whether that came across? On another day/another topic he could have done so much better. Or so much worse. That applies to them all.
Your DD sounds just lovely from what you have written about her and more importantly very involved in her subject and choice of college.
But others may have different experiences/advice.