I would be quite confident about top 20, cautiously optimistic about top 10, and any higher really depends on the reputation and strength of letters. If the writers are not known in the US this would pose a problem. But I am student, have never been on an admissions committee, so take all this with a grain of salt.
I wouldn't worry about the one 6 given the 7's in the math courses.
Also, apart from the weird cache around Oxbridge that still exists in Aus, why are you applying there? LSE is good (not sure about stipends) but I am not sure you can argue Oxbridge would be in the US top 20 or even top 30. Might still be a good fit, but don't apply just because it sounds fancy.