It depends somewhat on the course and uni in question. A "better" uni might not matter for some courses/careers (e.g. engineering, allied health professions, and some other similar things), and for others your "lower" overall ranked uni may actually be better for that particular course (for example while City is considered somewhat average overall they are extremely well regarded for their Business School).
Beyond that, if you would be happy on the course you've selected at the uni you're otherwise going to attend, then that may not be worth changing - doing a "lower ranked" course that you are content with and able to do your academic best in (and hence get a 2:1/1st) is better than going to a "higher ranked" uni and hating your course, the uni, the location, and struggling to even get a 2:2.
It can of course potentially be worthwhile; if you weren't really that keen on the course you applied to initially, have other reservations about the uni/living area, or if you want to go into a field where the "prestige" of your uni does matter (e.g. banking, law) then it might be worth considering. But I definitely wouldn't suggest automatically assuming you should go down that route...