Nothing, but if they don’t have time to prep for the SAT and won’t take it for that reason, they don’t have time to prepare for the TMUA either – and I don’t think you can just go hands in pockets and expect to get the requisite “scores”, even if you’re an AIME qualifier. (I do think this student could try and take the SAT with just a couple hours of prep but the TMUA is not in the same universe of tests.)
UEdinburgh, for instance, will accept AP 5s as qualifications for entry.