I know both universities well as an academic. THey're chalk & cheese in terms of size,. location, etc.
Lancaster isn't Russell Group but in many subjects it's highly respected and tip-top. And it's smaller, so students feel more "known" - Birmingham is fantastic, but very large & students can get a bit lost.
But then, Lancaster is in a small town & sometimes students can feel a bit claustrophobic - it's an hour from the nearest big city (Manchester or Liverpool). But only 40 minutes from the glorious Lake District.
The college system at Lancaster is not like the Oxford or Cambridge college - they're residential, social & pastoral - teaching is not done within the College, as in Oxbridge.
And there's a far higher preponderance of non-middle class people at Lancaster. Birmingham as a city is fabulously mixed and multi-cultural, but the University is a place of privilege - class/financial privilege shows up as educational advantage at about the age of 3 in this country, so most students at most universities which ask for high A Level results will have had substantial economic advantage.
I think that your DC really needs to see Lancaster - I'd live there in a heartbeat, but it's not for everyone. On the other hand, students can feel a bit overwhelmed by Birmingham.
And as for this:
It depends on the subject but generally a degree from Birmingham will be more highly regarded than one from Lancaster
Not in my discipline - Lancaster is as "highly regarded" or even more so, than Birmingham. Advice like this is not helpful - it really is discipline & specific degree-related. And "highly regarded" by whom?
Do you care to say the discipline? Then academics on here may be able to give you more specific information.