There's someone on this forum who got an offer from an Ivy League uni for a PhD after going to exactly one of those example universities. So it's certainly possible.
Also note that the Ivy League is a sporting association like BUCS. It has no influence on academics, it just so happens they're the oldest and hence richest universities in the US and are conventionally considered among the strongest academically - but there are plenty of non-Ivies that are as strong or stronger than Ivy League unis in general or in specific fields. For example Stanford, UChicago, MIT, CalTech, Carnegie Mellon, Berkeley, UCLA, etc.
Ivy League universities normally offer a full funding package for PhDs and generally it's expected in the US that you would only accept a PhD offer if it was fully funded. The criteria to get a fully funded PhD in the US is going to be similar to achieve the same in the UK - very good academic results (a 1st would be generally assumed at minimum), plus ideally research exposure if not actual experience, a relevant masters in hand would probably be helpful, good letters of recommendation from established academics who are familiar with your work, etc.
The main difference would be a research proposal is not usually part of the US PhD application process as I understand it, as you don't directly embark on the research project for PhDs in the US (unlike in the UK).
For a masters degree your options are considerably more limited as most of those universities don't typically offer a so-called "terminal masters" (i.e. a taught masters as in the UK). Many departments at those universities ?i.e. Ivy league and similar) in the US only offer a masters "en route" to the PhD and expect you to apply to the PhD programme itself. Funding is usually very limited for masters degrees, unless you get a TA/RA funded position or are a top level athlete eligible for athletics scholarships (I know someone who did that - I think she was an alternate for the UK Olympic team in her sport I believe though, the level is very high).