Anyway, it's not about who marries who, but who has children with who. First cousins in their sixties getting married - no problem. Gay first cousins getting married - no problem. Young man and young woman who are the result of several generations of cousin marriage in a relationship and having children - problem, but what can you do about it?
I'd start by making first cousin marriage/sex illegal. That's the first step towards making it socially unacceptable. You'd have to turn a blind eye to those already in relationship with cousin and not prosecute anyone grassed by their enemies, but if the purpose of the law is about the children that's doable, disabled children become the evidence. So if you've married your cousin, no more children for you after the law is passed unless you adopt. There'd be those that risk pregnancy and get away with it, without a disabled child nobody would know because the parents being cousins wouldn't come to the authorities attention. So those would slip through the net.
If marriage to cousin is illegal, you could test immigration applicants and decline married cousins. That would cut down numbers a bit and send the message that UK doesn't want cousin sex/marriage here. Those with genetically disabled children and where parents are cousins, even if not married, the children are the evidence it's a romantic relationship so the parents can't claim they're just cousins, immigration denied. This is due to the potential additional costs to the country of NHS and social care for these children. We have to protect the people already here who don't have enough resources as it is.
If public funds and prison space allowed, I'd start testing the parents of genetically disabled children when those children are diagnosed, to see if the parents are cousins. Then prosecute them twofold if they are, first for breaking the no sex/marriage to your cousin law and second for child cruelty. Because the child probably wouldn't be disabled if the parents had procreated with someone else, so the parents have chosen to deliberately risk causing harm to their child by inbreeding. If the disabled child's parents aren't cousins then it's just bad luck so no prosecution.
A couple generations of this disapproval and consequences and the practice of cousin marriage would soon die out.