I can tell you about Harvard. Apparently the Yale system is very, very similar. (I am involved at a low-level in interviewing UK applicants.) Harvard interview everyone who applies, as they feel a lot can be missed otherwise.
Everyone I've interviewed has applied to at least one other US college, as well as Oxford or Cambridge and another UK Uni. As well as top grades in GCSE's and A'levels, the student also needs to do very well with the SATS tests. A certain amount of leniency is applied to the Verbal Reasoning mark for UK applicants though, as this style of test is something that our students are just not used to, so they often underperform a bit.
I'm not involved in making final decisions about students - my report on their interview is one thing that the Panel considering, along with grades, extra-curricula and admission essays. Their ultimate consideration with UK applicants of a high caliber, is; Does this student gain something from coming to Harvard that they could not gain from Oxford/Cambridge? Many excellent applicants are turned down purely on this factor; a deep commitment to the Liberal Arts system on offer is something that should be demonstrated to help avoid this.
There are some things I've picked up along the way. These may not be set in stone, but are things I've joined the dots on myself.
- No student I've interviewed has got a place if they've got even 1 C (or below) at GCSE. I've had amazing candidates with 12 A*, but then a C in art or a language they've suddenly picked up for a bit of fun; it doesn't seem to matter, its been a no.
- The essays are key in standing out from the crowd. These need to be erudite, intimate and unique.
- The level of extra-curriculas is important. Basically, the Panel want to see that something has been taken to the highest level it could be, according to the opportunity the student has had. Grade 6 piano and on the school A netball team doesn't cut it. Harvard has amazing sports coaches, and sometimes they will lobby the Panel to "get" a student perfoming at a high level in a Varsity Sport. It can be worth contacting the coaches directly about this - the interviewer will discuss this with applicants.
- Someone from a family & school where there have been less opportunity for extra-curricula are given more credit for lesser achievements. However, the problem these student then face is getting over the aforementioned hurdle of "what would you get out of Harvard that you personally couldn't get from Oxford". A massively higher proportion of Independent school children still get in, though not exclusively so, and we're working on it! Last year was a bumper year for Eton boys getting into Harvard, for example.
I can answer questions about the interview process if that's helpful for anyone.
I'm a big fan of the Liberal Arts system, and would love our DC to have the same opportunity; we specialise way to young, in general, in the UK. Harvard is an amazing experience in so many ways, but its highly oversubscribed and there are other great colleges and Universities too.
HTH