Level of Competition at UK Econ Masters

I'm an undergraduate student at a top 20 American liberal arts college. I'm graduating next year with a degree in applied math and economics. I'm quite interested in pursuing an additional degree at a UK-based economics masters program and my dream programs are LSE and Oxford. However, I know it's quite competitive and am curious of the extent of the competition and what the benchmarks are and have traditionally been for highly successful applicants.

If anybody has thoughts on my specific situation, I have a 3.97 GPA, I'm a winner of an economics departmental award (contributions to the department), in roughly the top 5% of my overall class (college-wide), interning at a top (AG, Cornerstone) Econ consulting firm, currently leading a research project with a professor that I hope to pursue publishing my senior year, and have a long history of being a teachers assistant in the mathematics department (most recently on the proofs-based mathematical reasoning course). I haven't officially taken the GRE but anticipate scoring in the 95+ percentile (based roughly on a 99th percentile SAT score).

I do have some weak spots too. My leadership in extracurriculars is minimal, and in part my GPA is propped up by A+s in courses (mostly econ); I have a roughly even mix of As and A-s in my math courses and got an A- in microeconomic theory and Econometrics (and, annoyingly, an intro stats course). Does anybody have a clue what the benchmarks are for a competitive application and what would likely make the cut?