Oxford MFE vs MIT MFin for MBB Consulting Firms

It honestly doesn't matter at all. Both are fantastic programs that will set you up for invites by MBB, if the rest of your profile is on par as well of course.

However, there is a huge difference in location/recruiting. If you don´t have a visa/are a citizen, it is almost impossible to start out in the US. Also consider that atleast M does their recruiting by country, and you have to redo the hiring process if you want to switch before starting out (e.g. after a successful internship). In Europe the MIT MFIn is very well regarded as well, so you could easily interview from there for an European spot. Some US based monkey should chime in about Oxfords reputation in US offices, but I would assume it´s good as well. E.g. the DACH offices of MBB also have a recruiting weekend in Boston every year, so you could easily do that from MIT, without having to fly across the pond all the time.

So I would rather base the decision on what program interests you more and really consider which office you want to apply for.