I also wish the DD were doing Maths, @PermanentTemporary However I assume that where it is not required the programme compensates for this.
OP, sadly CS at B’ham requires Maths so that’s out. However Lancaster. one of the hidden gems of UK higher education, does not. Nor does Cardiff (or Swansea, of course) or Leicester.
Cardiff is Russell Group and Lancaster, which has come on very strong in terms of both research and Teaching and Learning recently, is referred to as ‘Russell Group +’ together with Bath. The Russell Group is the elite group your mum is probably thinking of. Lancaster and Bath are now acknowledged to be stronger than many RG unis, hence the +.
(As an RG academic I personally think this is a load of nonsense. )
Gently, I wish you would consider Cardiff and Lancaster for DD even though they may be further away. Maybe Leicester. (Unless she needs to live at home, of course. Reality is that some need to live with financial constraints and this can be done with dignity.)
I know, she’ll be 18 or 19. She seems so young and some unis seem so far. But one of the big advantages DC of parents who went to uni have is that those parents are less nervous about sending their DC away! DD like all DC deserves to find her best fit if finances permit. Hopefully she will soon be able to get well paid summer work.
Anyway, the suggestions you’ve gathered here plus your original ideas should give DD plenty to go on with. I am very sorry her CS teacher didn’t help her. Now DD and perhaps you and/or DF should attend some Open Days to see which places appeal to her. They are happening now!
Learning about each uni’s contextual offers is indeed a real slog. I am not sure you mentioned it but some use Free School Meals as a flag. Also the high school or college post code. You can ask the admissions tutors at OPen Days about contextual offers, also. Actually they should be discussed as a point information