soulgroove thanks for your great reply. It was really helpful. Let me clarify some stuff about me:
I already have two MSc (one from LSE), so I´ve had plenty of class work, I have already a clearly defined research proposal and significant work experience. I did not apply directly to the DPhil because many people told me it was fairly difficult to go in unless I already had a supervisor ex-ante, so I went for the MPhil.
Anyway, my take at this point is to move as fast as possible to the PhD as possible (no more class work!), and LSE gives a single year of MPhil plus no re application to move to the PhD except getting a 65, while Oxford gives me two years of MPhil, a re application process that has uncertainty and administrative work. By the way how engrossing is the administrative work? At this point I am more keen on asking to move to the MSc research at Oxford to move faster to the DPhil. However I am still open to suggestions...
2) I think I have a fairly good supervisor at LSE, however do you have any suggestions in Oxford? I like quantitative comparative politics, econometrics and political economy. Also I wanted to ask why didn´t you end up finishing your DPhil at oxford? (If its ok to ask).
Thanks a lot for your reply! Very helpful!