Approximately 3% of UK births are to parents of Pakistani origin. They account for a third or so (see below) of the total number of children born with genetic disorders. Over half of people from the Pakistani community are married to first cousins (or closer).
These are not controversial data and a simple Google search will generate much reading material.
I work in medical genetics in a city with a large Muslim community. We have a very active clinic.
The absolute risk of genetic disorders in children from first cousins is around 4% (and this is the figure we quote to first cousin couples who have no history of any genetic disorder). The background level in the UK is around 2%. Therefore, the actual risk - 4% - is not huge, but the increase in risk is 100%. So, we see varying reports of how serious an issue it is.
There is evidence to suggest that the incidence of genetic disorders in the children of consanguineous parents is underestimated, particularly those which are difficult to define as a specific genetic disorder e.g. a learning disability. This is thought to be for two reasons. First, is somewhat hidden by the socioeconomic benefits of marrying a first cousin, such that one can more easily afford things like extra learning support and private tuition. Secondly, in the absence of severe physical symptoms (and sadly, sometimes in the presence of these), there can be a tendency for family groups to close ranks. When clinicians go out into the community, they uncover far more children with genetic abnormalities than they have "on the books" at a hospital unit.
First cousins share 12.5% of their DNA. This is an odd stat to quote because obviously, they share closer to 100% of their DNA as human beings. The 12.5% figure represents the DNA that comes from the same ancestral source, rather than just being coincidentally the same because we are the same species. Anyway, this means that children of those parents have approx. 6.25% of their DNA that is from the same ancestral source. This is a huge amount. It zings out at us when we analyse DNA samples. It's unmistakeable and, to be honest, quite conceptually shocking. And it's the source of recessive genetic disorders (contrary to PPs above, the incidence of dominant genetic disorders in these children - where one parent may have passed a dodgy gene down - isn't any higher, IIRC).
First cousin marriage, from a clinical genetics POV, has underpinned the entire discipline for decades. There is no way we would have even half the knowledge we have now if it weren't for first cousin marriage.
But that's not really a good reason to allow it to continue. In my genetics view, first cousins are a little too close for children. As a liberal, I wouldn't stop anyone marrying a first cousin if they wanted to (but perhaps that isn't quite the case in these communities). As a pragmatist, I doubt you could stop it anyway - people would lie about family relationships (although, as I say, it would become apparent the first time we see their DNA). I have spoken personally to various people within the community and the risk is becoming more and more a topic of open discussion, and I think that's the right way forward - information, education and allow people to adapt accordingly.