When you say "postgraduate studies", do you mean a masters, or a PhD? For the latter certainly, "brand name" is meaningless compared to the renown of a) your supervisor b) your research group and c) the department itself overall, in descending order of "importance" in academia. Some of the best departments for PhD studies oin a particular niche area of subject are at universities which may not be considered that "good" by students looking at undergraduate degrees. For example, Sheffield is considered one of the, if not the, best universities in the world for feminist philosophy, by philosophers actively working in academia. Many undergrad applicants may not consider it their primary choice for an undergrad degree in philosophy compared to e.g. Oxbridge, LSE, UCL, etc.
For a masters degree, it really depends on the type of masters (taught or research) and what you plan to do with it afterwards. For research masters, or taught masters where you're planning to go on to a PhD, the above is probably going to be the case. For going into work, then depending on the particular sector and work you are seeking generic "brand name" appeal of universities may not be as relevant. Often clinical roles in the NHS require you to have taken specifically accredited degrees (at the UG or PG levels), which are rarely available at "brand name" universities - this doesn't disadvantage their graduates when they are the only ones who are able to apply to those roles directly (for example, to band 5 biomedical scientist roles in the NHS, where you need HCPC registration and an IBMS accredited degree to apply to them directly after graduating - something that no Russell Group university undergraduate BMS degree offers currently).
Likewise for some specialised fields otherwise, certain programmes may be well known and established at the masters level; for example, most curators of major museums and galleries in the UK have (historically at least) done the MA at the Courtauld Institute, quite possibly over the art history programmes at Oxford or Cambridge. This particular programme spawned many of the current industry leaders (well, maybe a little less these days, but this is just an illustrative example) and so by connection with alumni offers a huge amount of industry connection and networking opportunities that may not be available from other programmes considered academically strong otherwise.
Essentially, things aren't so cut and dry at the graduate level, as may be seen to be at the undergrad level (I would argue they aren't so cut and dry at the UG level either personally). Without a lot more information about your plans it's a bit hard to make suggestions, other than to apply to all of them if you can and then make a decision once you know which you have offers from.