Not defending cousin incest, but it sounds like the NHS is at least backing up its viewpoint with evidence.
Now as to unstigmatising cousin marriages, that's a no from me. There are 60 million other people in the UK, there's gotta be at least one that's right for you that's not also your cousin.
P.s. Trump should really have left the US out of this conversation given how infamous some of the Southern States are for this sort of "matrimony"