You will need SATs or ACTs for essentially all of the most competitive schools. These cover GCSE level maths and english. You do not get better at this stuff the longer you are away from it. If you are seriously thinking about an American university, I would take these in the Spring of my lower sixth. Most of the top schools “superscore”, which meanss that if you take the exams more than once, they will only consider your top score. If you can afford it, I would recommend taking them early, and if you need to, you can always take them again later.
I disagree that you need to convince the schools that you are “well rounded”. While that does not hurt, the top schools are not after all well-rounded students. They are after a well-rounded class. That being said, with limited space in each class, they want to take students who will add to the life on campus. Consider two students with equivalent grades and scores. One spends all of his/her time in their room alone, studying. They other started a band in primary school, then got to secondary school and started a band, then got to sixth form college and started a band. There is a pattern here. Given equivalent academics, which do you think would be more fun to have on campus?
The competition at the top schools is so fierce that there isn’t really an unimportant part of the application. Grades, Scores, Letters of Recommendation, Essays, Interview. There is not one thing that can get you in (or usually keep you out), but you really would not want to screw up any part of that.
Also, the UCAS personal statement is totally unusable for virtually any US application. They are very different documents that serve very different purposes. For UCAS you want a paean to your subject. If you want to read Physics, you need an essay extolling your love of Physics. For a top US university, that is almost never what they are asking you about.
For impartial advice on US universities and admissions, I recommend (not for the first time on this board) the non-profit, independent Fulbright Commission, and their advice service. Awesome, and without any agenda excepting to encourage Americans to study in Britain, and vice versa.