Hi.
The question in the title was: "How common were the Magdalene Laundries in the UK?"
They existed in Ireland. I don't think any existed in any other parts of the UK. The first seems to have been established in 1765 but was unusual in being a protestant organisation. Most of the Magdalene laundries were Roman Catholic and some of these seem to have continued until very recently: Wikipedia suggest some time in the 1970's but an article by the History channel suggests it is much later than that: The history channel claim that the Gloucester Street Laundry was home to 40 women and did not close until 1996. Both sources estimate that 30,000 women were confined in these institutions in total.
I do not know exactly how many such laundries existed, I have the impression they would have been in only some of the city areas of Ireland. A full map or record of all the laundries is probably not available:
Estimates of the number of women who went through Irish Magdalene laundries vary, and most religious orders have refused to provide archival information for investigators and historians.
[Source: History channel website. https://www.history.com/news/magdalene-laundry-ireland-asylum-abuse]
What confuses me is how they were able to function just decades ago, despite the UK's Slavery Abolition Act of 1833. Were the metaphorical bars of these prisons in the form of the girls having no money, and basically having the misguided understanding that no would else will hire or care for them in any way? And perhaps also that the facilities were the pill they had to swallow to get into heaven?
I'm sure those were relevant factors. The Histroy channel article adds this comment:
How did such an abusive system endure for 231 years in Ireland? To start with, any talk of harsh treatment at the Magdalene laundries and mothers’ homes tended to be dismissed by the public, since the institutions were run by religious orders. Survivors who told others what they had been through were often shamed or ignored. Other women were too embarrassed to talk about their past and never told anyone about their experiences.
I should also add that some of the women put in the care of the Magdalene laundries are described as orphans and victims of abuse, so that they may not have had any family members they could complain to or that would have listened to them. Some of the women in the care of the Magdalene laundries were imported or routed to the laundries from psychiatric institutions and/or are described as having "special needs" and may not even have been able to effectively communicate their experiences to the outside. Finally, an unknown number of women died in the Laundries and they would obviously not go on to describe their experiences:
....the cemetery at Donnybrook was no ordinary resting place: It was a mass grave. Inside were the bodies of scores of unknown women: the undocumented, uncared-about inmates of one of Ireland’s notorious Magdalene laundries....
[History channel, as above]
"Best Wishes" hardly seems an appropriate way to end this post.