My guess is that up to the 1960s there were lots of places like St Margarets that took in unmarried girls and gave their babies to people by-passing the official system. The girls were probably sent by their employers or GPs. Whether any money exchanged hands I have no idea.
When I got pregnant in 1959, my dad spoke to a friend of his who was a GP and he found such a place for me which was a maternity home. Many of the pregnant girls had to live in and worked as assistants (skivvies) and after their babies were born, they were "given" at 10 days old to couples who wanted babies. I was more fortunate, in that there were no places left for me to work in the home at the time, so I went to stay with a lady and her two young children. She was getting divorced (a famous at the time TV personality) and was happy for the company. I stayed 10 days after I gave birth and then walked out without the baby, who was picked up the same day by her adoptive parents. There was non of the 6+ weeks looking after a baby before giving it up for adoption.
When I went to sign adoption papers a few weeks later I was told not to say anything to the officials (whoever they were, ?social workers) about what had happened, just that I agreed to the adoption.
For years I worried about why a couple would not be able to adopt by the official channels, were they too old or something. However, I've since met the adoptive couple (and my daughter) and they were not too old or anything, just a normal couple who had been married for about 7 years and hadn't had a child. They told me that someone had told them about the home and that they could ask for a baby and just be given one

However, they went straight to social services (or the 1960s equivalent) the following morning to tell them about the baby and worried until I signed the papers.
Lizzie