You are in a not dissimilar position to ours a couple of years ago - we didn’t see the point of paying $300K for college, but three years at Oxbridge was half that and not much more than our in-state options. Since both of us went to Cambridge for undergrad and PhDs (and our kids have dual citizenship) it was natural to consider the UK as an option. So I would consider this to be a very viable proposition (though unfortunately my S didn’t get in, he may consider going there for grad school), but don’t underestimate how different the UK system is and how much you have to love your subject to be happy there (and just to get in). You also have to be really good at hard exams, because that is almost all of your grade (at Cambridge the whole year came down to four 3 hour exams over 2 days).
A few things to consider:
- Oxford is generally much more favorably disposed to Americans than Cambridge, it admits more of them and the MAT (in early November pre-interview) is much better timed than STEP (in June so you don’t have a result until August by which time most US colleges have already started their fall semester - the most realistic option, although only possible for a very advanced student, is to take STEP at the end of the junior year of high school).
- The cost differs by subject: math is substantially cheaper than math+CS (by about $10,000 per year). Also Oxford is cheaper than Cambridge (by about $5000 per year).
- At least in the UK (and some other places like Singapore), Cambridge is viewed as more prestigious than Oxford for maths, but conversely, Oxford is better known and perhaps more respected than Cambridge in the US.
So although I did maths at Cambridge I would generally recommend most US students to focus on Oxford for undergrad (see my comments here: http://talk.collegeconfidential.com/united-kingdom/2091170-oxbridge-admissions-for-americans.html). Then you only need a minimum of three 5s in related subjects, which makes life more straightforward (though if you got a top STEP score in junior year of high school then I very much doubt any offer from Cambridge would be conditional on senior year results).
STEP is really tough and most colleges prefer STEP 2 and 3, not STEP 1 and 2 (you usually only do STEP 1 if you haven’t taken Further Maths A level, and the vast majority of students do have Further Maths). The level of preparation in the UK (and places like the Far East and Eastern Europe which contribute most overseas students) is very significant, because you specialize in maths for two of your three or four A levels, and the top schools which contribute many of the most able students have very strong teaching. As an example, in school I spent two years doing 3+ hours of maths per day for A levels, with three different math teachers (plus physics on top) who all had Oxbridge math degrees or PhDs. 5 of the 10 kids in my math class went to Oxbridge, 4 got firsts. Difficult problem solving was emphasized, not plug and chug at all, and exams are really long tail - usually 70% is an A in the UK system.
Don’t worry too much about AP stats, take MVC once you’ve done Calc BC. If you get to Linear Algebra then you’ll be ahead (although everyone does the same set of compulsory courses in the first year, and it is very heavily proof based from day one). And perhaps study the Further Maths syllabus (but there’s no need to take the exam). If you look at the STEP exam you’ll see it is heavily weighted towards pure maths questions, which is fairly representative of the UK A level coursework (and the majority of the strongest Cambridge students ended up focusing on pure maths). The STEP exams (including the marking scheme) are almost identical to the structure of Cambridge undergrad math exams.
The thing that will impress the most is getting beyond AIME: IMO qualification is essentially an automatic admit (see the article in here: https://share.trin.cam.ac.uk/sites/public/Alumni/The_Fountain_Issue_19.pdf) and AMO qualification would be a really big boost. Other math competitions and math circles would be helpful too.
@HazeGrey has a son at Oxford doing math+CS and I’m sure will chime in.