Marriages between first cousins could be banned in UK under new proposal | The Sun
Proposed bill to ban cousin marriage. Cousin marriage is already (gradually) falling out of favor among British Asians, but I have no issue with banning it altogether.
In terms of "When did it become unacceptable?" the answer is "In the early Middle Ages, and then once again in the 20th century." The early church in most parts of Europe banned cousin marriages even up to very distant relations - at its peak, church policy banned even marriages between sixth cousins, which must have made finding spouses quite challenging. Rich people could often get round the rules by paying huge dispensations to the church.
The purpose of these rules was not about genetic health, but about limiting the wealth of powerful, wealthy clans. Until Christian times, Europe (like everywhere else) used cousin marriage as a way of cementing clan loyalties and building up lots of wealth and power within particular families and tribes. The church did not like this, as its policy was to limit the power of secular groups to ensure that the church reigned supreme.
When the Protestant reformation took place, the new Protestant churches often got rid of the no-cousin-marriage rules; I'm not sure why, but perhaps they put these rules into the whole category of "the church has too much power and imposes lots of supersitious rules as a way of forcing people to pay dispensations." But by then, the "work" of centuries of cousin-bans had already been done; even without the rules being in place, Europeans were now largely in the habit of looking outside their family group for a spouse, so cousin marriage remained fairly rare by international standards. You did get the odd cousin marriage, but it was nothing like the situaiton you saw in other parts of the world, where often 50% of more of marriages were between cousins.
The fact that Europe banned cousin marriage so early is believed to be one reason why Europe developed modern, individualistic ideas and the Renaissance and the Enlightenment and so on, rather than other parts of the world, because young men and women were forced to look further afield for spouses, the average age of marriage for women rose and women started to exercise more autonomy their choice of partners and so on, and all this encouraged a more individualistic approach to life where you saw yourself as a citizen with free will, rather than a member of a particular clan or tribe.
For all these reasons, I'd be strongly in favor of banning cousin marriage. Between the Reformation and the 20th century, we could "get away with" cousin marriage being technically legal, as few people wanted to do it anyway. It's an issue with some ethnic groups, who are using this as a way to practice "chain migration" of (often uinskilled) relatives from poor countries, which is not an example of good/beneficial immigration and risks fuelling fury and a backlash among everyone else. And cousin marriage is inherently not compatible with being a rich, modern liberal society.
That's even before we get onto the genetic stuff. And the pressure this puts on young women of Asian background in this country, who (despite being brought up in a liberal society where women have many opportunities and choices) may be pressured into marriages with male cousins from poor countries who have predictable attitudes to women and gender roles - and if they refuse, may end up with a limited choice of spouses, due to the fact that many men from their own social mileu may be happily pairing up with female cousins from the same poor countries, with "traditional" attitudes.