There's very little information about how people were screened and accepted as adopters. Obviously religion was an important consideration because most adoptions were organised by religious organisations. In the years up to the 1970s the unmarried mother was seen as a sinner, a fallen woman. It was important to make sure that her child was brought up in a household where it would learn right from wrong, which meant a religious household.
There were circumstances where a baby might be rejected for adoption. These included disabled and mixed race children and those whose mothers had mental health issues. Only 'perfect' children were elligible.
It's not possible to discover what criteria (apart from religion) were used to choose parents for these children. There were medicals and references were sought, I believe, but how much weight was given to them isn't known. I was adopted by a man with serious heart problems, but I don't know if that could be diagnosed easily in the 1940's. All I know is he only lasted two years after the adoption. The Bamber children were given to a woman with mental problems but we don't know why.
My adoption was arranged by the East Anglian Adoption Society in Cambridge. I don't think it was affiliated with any religion or church.
My birth mother said she had no interest in religion and the adopters could be Buddhist for all she cared! My adoptive father has always been an atheist and my adoptive mother agnostic.
My file shows various questions asked of my birth mother eg family history of physical and mental health and alcoholism. There's also a letter from her GP and various references but Interestingly now you've mentioned it there wasn't anything for my adoptive parents.
There seemed to be a lot of emphasis on 'matching' by way of physical appearance: height, weight, hair and eye colour. I've always thought the way these adoptions were arranged as on par with Nazism! Luckily I was described as a fine child but what if I had been unfine? I wonder what happened to any of the Thalomide babies put up for adoption?
I once had a boyfriend who was adopted and his adoptive father died in a car accident when he was 2 yoa.