Several kids from my local high school went to St Andrews and Edinburgh this year and for the last few years.
Students have gone to German universities this year too. Two are in the Universität der Künste Berlin, but I don't recall where the other two went.
There is no major exam akin to GCSEs, but US students operate in a system where their grade point average will be used by universities to sift and sort them when they apply. Assessing the GPA begins with your first assignment in your first week of high school. It takes an incredible amount of consistent slog to get a perfect GPA (4.0). Some school weight honors and AL coursework so a student could end up with a GPA somewhere between 4.0 and 5.
My DCs spent their time juggling sports practices and meets, participation in school musicals, and about 4-5 hours of homework and studying nightly.
On top of the GPA and the consistent work that securing a decent one entails there is a raft of standardised tests that US students take. There is the PSAT, NMSQT, SAT, ACT, and SAT subject tests along with AP coursework and exams. All of these are administered on a nationwide basis, so you can really see where you stand when you get your results. Excellent results in these are necessary for admission to selective US universities.
It might be difficult for a UK student to get the hang of the format of most of these tests with the exception of AP exams. Some measure thinking and reading comprehension skills while some measure sunbect knowledge. A UK student would need to go to prep courses to get up to speed on strategy.
What if we only stayed 2 years (which would be equivalent of GCSE years) then returned to the UK, would they only get half a high school diploma or nothing?
Your student's courses and coursework from their UK school would be assessed and grades would be translated into US equivalents.
Most high schools and all states have graduation requirements that must be met if a student is to graduate. These are bare minimum standards, not something all students aspire to. Examples are Driver's Ed, two semesters of swimming, four semesters of humanities, six semesters of English, six semesters of math, two semesters of science, two semesters of consumer ed, and so on.
It's entirely possible for a UK student to graduate from a US high school with a cobbled together academic record. Chances of getting into a good university would depend on caliber of courses taken in the US. Honours level and AP coursework plus SAT subject tests, and good grades in these classes/exams would be necessary.
Most US universities require much more than the minimum. To have a shot at getting into a highly selective university in the US a student would need a GPA higher than 4.0, and then there are the extra curricular expectations.