If you want to do an exchange at a US university to build contacts for possible postgraduate study in math, then you'll want to remove programs that are not known for strong graduate programs in this area, or which primarily are undergraduate schools. These include:
Edinburgh: Babson College; Haverford College; University of Richmond
Glasgow:Radford University, Virginia; University of Alaska Fairbanks; University of South Florida; University of St. Thomas; University of Wisconsin, Eau Clare, La Crosse and Stout UW-Madison--very good, though).
I don't know about the others.
Your lists include a fairly diverse range of universities. You also should consider what geographic locations region, urban/rural, climate, etc.) and other features of the university might appeal/not appeal to you.