There's no reason you couldn't get in on the basis of being from the UK. You will however need to get excellent grades across the board and as noted, from year 10 onwards they will want to see you engage in extracurricular activities across a range of areas, at a very high level of achievement and in leadership positions. Just showing up to debate club once a week wouldn't be suitable for example - they would want to see you captain your debate team to a regional/county/national championship for example. Likewise just playing X sport or instrument by itself isn't usually sufficient, as they want to see you have engaged in leadership roles through that and distinguished yourself on at least a regional level.
This in contrast to the UK system where it's usually sufficient to demonstrate excellent grades in just relevant subjects to your proposed area of study, and where extracurricular activities are usually seen more as a bonus and aren't specifically required or expected (and where for Oxford and Cambridge they are only really interested in activities you've done specifically related to your proposed area of study). Also bear in mind in the US you don't apply to a specific subject of study from the start, you just apply to the university "in general" and then when admitted you are required to take a range of courses across different areas before declaring your major (and often still take some outside of your field after declaring your major). Note even if you know what you want to study you still need to take these courses (called general education requirements). So if you want to study nothing but computer science and maths, in the US you will also need to take classes in literature, history, social sciences etc.
Note that most of the Ivies to my knowledge are commited to meeting all demonstrated financial need, and several (if not all) are need blind, which means a) they will make offers without regard for how much they would need to pay to get you to study there and b) they will cover all costs needed for you to study there based on their calculation beyond your ability to afford it. That said it's worth noting for b) they will calculate a "family contribution" and you will be expected to pay that amount of it yourself (this is based on your family income). However they generally cover the rest. So finance wise there is less of an issue aside from travel costs (potentially also other extraneous costs too, like health insurance as there is no state/national healthcare in the US).