Does anyone know how competitive admissions to these programs are? I need to improve my background for doctoral applications and the best U.S. departments don't offer master's degree programs. I'm an American and don't know much about the British educational system so I'm pretty lost.

I'd be applying to the 2-year programs at LSE and Oxford. At Cambridge I'd officially be applying for the "Diploma" but from the website it sounds like the Diploma and then a one-year master's is functionally the same as a two-year master's program.

I definitely meet the minimum qualifications described on their websites (I graduated from a top U.S. institution with what's roughly the equivalent of first-class honors) but I haven't been able to find any information on things like acceptance rates.