I work woth and teach medical genetics in the urban NW, at a national genetics centre. Assuming the OP is addressing the child issue and not the marriage issue, this subject is pretty much the topic of every ethics talk we see
Some thoughts...
Some of the family pedigrees (relationship trees) I have seen have taken days to untangle. And illegal or not, uncle/niece (less frequently aunt/nephew) don't even raise eyebrows any more.
We see some families who are very engaged in the whole process, from whom we assume complete disclosure about familial relationships in a clinical setting. However, we also see families who refuse to engage, who have to be visited at home, who evoke gods/demons/curses/etc as explanations. Regardless, the expectation is that making first cousin marriage illegal in the U.K. would do nothing but create lies about true familial relationships, in order to evade bans.
For first cousins having children, the risk of congenital defects are relatively higher than for non-consanguineous parents, but not really absolutely high. The stats are heavily skewed so that the British Pakistani population creates a vastly disproportionate burden on the NHS in medical genetics.
Having said that, there is a suspicion in the genetics community that the figures for genetic birth defects from first cousin marriages are artificially low. When we consider:
- The most common congenital issues include 'generic' developmental delay.
- A culture that promotes first cousin marriage can often promote secrecy/hiding problems.
- A culture that promotes first cousin marriage is often doing so to protect family interests ('money') and they often have the means and motivation to minimise issues like developmental delay (paying for good schooling etc).
It's more straightforward to identify a child delayed from average to 'problematic', near impossible to identify a child delayed from brilliant to average. The latter is unlikely to represent a clinical burden, of course, but that doesn't mean the parental genetic relationship has had no effect (so interesting from an academic POV).