I hope this reply is in time to be useful (since their application rounds run in the autumn, it should be).
There are some programmes at Oxford and Cambridge for which relevant professional experience and motivation might more than make up for a poor performance in an undergrad degree, especially given that that degree isn't even your most recent qualification. These tend to be slightly more vocational courses. In an application, you could point to your law school performance as proof that you can study effectively, and also as evidence that your academic performance has been on an upward trajectory. Your best bet, I think, would be to email the graduate admissions office who cover whichever programme you're interested in, and ask them.
The UK government is increasingly pushing on access & diversity in British universities, and plenty of individual people and institutional facets within Oxford and Cambridge are concerned about minority access. However, at the moment that government action is very much focused on undergraduate admissions and, within undergraduate admissions, on applicants who are resident in the UK. Large-scale institutional thinking about access hasn't really reached international students or graduate students yet, though it probably should.