Hi, OP -
I am a former RG STEM admissions tutor and DH is an emeritus Maths professor, also RG.
Firstly, very few unis require admissions tests. However that list is slowly growing. I cannot recall a School using such a test going into Clearing, and I am not just thinking of Maths. This worry is likely to be a red herring.
The suggestions you’ve been given above are for the most part excellent. Amongst them, Lancaster and Birmingham have really come on in recent years. Lancaster is now a UK Top 10 in research and has put a lot into Teaching and Learning recently. Birmingham has always had quiet strengths, partly complementing its outstanding Computer Science, so it is nice that recognition is now coming.
I don’t think there is a bad one in the lot, though I don’t think of STEM as one of Exeter’s particular strengths. The others defo have solid maths.
Does DS have FM? If not, I hope he will give some thought to choosing a programme where the majority also have not had it. When a significant majority have had FM, the Teaching and Learning style inevitably accommodates them. Those without FM lose out - in theory they get all the background they need, but it takes a combination of very high ability, very high confidence, and a lot of work to process it all properly. We get many posts on the FE boards from Maths and STEM alumnae who never found their feet because of this.
If FM is not a barrier, I agree with everyone who says it is worth attempting an internal transfer at the current uni, which would prefer to retain DS’ tuition fees (assuming this is desirable). When I was an admissions tutor we made these trades frequently. Economics is so competitive that unless there is an admissions test for Maths that DS lacks (or FM is a barrier) or policy is very rigid, I would expect chances to be high that he satisfies the Maths admissions requirements.
If this is not feasible or does not appeal, then it would be great if DS could visit a few places if he has not already done so. This could do a lot to inform his choices.
Best wishes to him