I believe the reason that Buckingham will not show up on some ranking tables is that, unlike almost every other university in the UK, it is an independent and private university. As it does receive state funding it also does not undergo the assessments which are often used to determine and rank university quality, such as RAE.
I don't really know much about the university from a student perspective, or what difference being a private university will make (apart from in terms of cost) I'm afraid.