EM oxford or LSE econ?

What kind of course are you looking for? I expect the LSE course will be more mathematically involved than the Oxford one (or at the least you will have more opportunities to take more mathematically rigorous options through it). The LSE course also permits you to take outside options (which can be to take more mathematically rigorous options in the maths or stats departments, but could also take totally unrelated options of personal interest in e.g. economic history, philosophy, anthropology etc).
In terms of examination formats bear in mind the Oxford exam format is that you take a set of exams at the end of your first year (which I don't believe count towards your final classification) then all the remainder of your exams are taken at the end of your third year (so no summative exams in second year normally as I understand for EM); this means your entire degree classification (aside from dissertation or any papers assessed by submitted work) hinges on your performance in a ~2 week exam period at the end of three years. So that could be quite a pressured experience.
Conversely at LSE you take exams I understand normally at the end of each year in the summer (some options may have January exams too?) and each year counts somewhat towards your final classification (with much less weighting on first year, moderate weighting on second year, and heaviest weighting on third year as I understand). This means you do need to remain consistently engaged throughout the course (although one would hope you would otherwise), but also might lessen some of the pressure of having everything ride on a set of exams at the end of the three years.
Another big factor is living costs - London is extraordinarily expensive and it's hard to escape that fact. If you come from a low income background you can be eligible for bursaries at either uni (and at LSE potentially a rent reduction when living in halls; I think some Oxford colleges have similar offerings), but it's still something to bear in mind. Also consider that at LSE it's likely you would only be living in halls in first year and for second/third year you would be living in a private rental in London (which might be somewhat farther out from where the uni is - you could have a moderate tube commute each day!). At Oxford the exact format varies a little between colleges but for most colleges you live in college in first and third years, many also for intermediate years, and even those where you live "out" in second year this may still be in college managed accommodation. While Oxford is also an expensive city to live in and the college rents are a fair amount for a student, relative to the private rental market I gather they are competitive.
Obviously also the experience of living in London vs Oxford is going to be very different, even aside from considering "college life" at Oxford they're just going to be very different cities. Also bear in mind Oxford has shorter terms and thus longer breaks between terms; although often I understand there is set work between terms and formative assessments when you return after the breaks, this could give some flexibility for e.g. pursuing a short term (ideally paid) internship for example, or simply saving money by being at home for longer.
In terms of career prospects I doubt there is much between the two, and suspect it would depend more on what you yourself push yourself towards at either institution.