Basically, for my chosen career route, I need to have a postgrad degree. Therefore, does it matter where I do my undergraduate studies?
Not necessarily as long as the classification is good.
E.g. My philosophy teacher went from Greenwich to UCL o.O
There's a limit to how much universities care about where you did your undergrad, but employers still might. All other things being equal, you'll never lose out from going to a better uni though.
What degree subject / career path do you intend to follow? It can matter in some subjects more than others.