Glad she's on the mend, AH.
Whistl as far as maths hierarchy goes, we didn't include northern unis but DS1 says Cambridge over Oxford, then Warwick, UCL then Bath. Then Bristol, then Exeter, Southampton pretty similar then Nottingham and Birmingham. After that Cardiff then maybe Loughborough, Leicester, Reading. We didn't really look at London either.
I'm sure Durham, Manchester, Sheffield and Newcastle are up there but they were too far away for us to consider.
DS1 was high ability and did STEP but wasn't quite Cambridge standard. 3 x A* not good enough alone for Cambridge! Your DS's school does send a few to Cambridge, though and supports STEP so your DS should get advice from school on which unis to look at.
My DS3 doesn't have DS1's drive, however, so we'll be looking at a different selection of unis for him in all probability. He's going to find out if the school supports STEP as even if you don't pass, the study alone helps with all your other maths.
STEP 1 is aimed at maths A level students, STEP 2 the same but harder and STEP 3 is hard FM. Some unis include STEP in their offers, e.g., if you get any STEP at grade 1 you can drop a grade in FM etc at Warwick or any STEP grade 2 can drop a grade in FM at Southampton I think. Mind you, DS1 found getting an A* in FM easier than a grade 1 at STEP. But doing STEP made his maths better.