@mdpmdp @Twoin18 has already hit the important stuff (and a lot more). Just adding my perspective:
My son chose Oxford over Cambridge because of the MAT vs. STEP timing difference and the lack of a true joint degree program.
His AP path was similar - CS A in 10th, BC & Physics C in 11th. That was all Oxford cared about and I believe it was helpful for those to be completed at the time of application.
My son was a three time AIME qualifier and on our state’s ARML team for three years, but he never got to the JMO/MO level. Senior year he did do a math circle that competed at HMMT/PuMAC, but that was after application.
At Oxford, the joint Maths/CS program is a four year program, so not as much savings there, plus as was noted, that is one of the more expensive degree programs. The three year straight maths is a more cost effective option.
My son’s experience so far - prelims and second year exams have been quite challenging. Way above and beyond the MAT.
Please feel free to DM with any specific questions.