I'm well aware of the history, I am the product of exactly what you've just described but having said that since the Adoption Act of 1925 there's no such thing as an illegal adoption (in theory). All adoptions since then were conducted as per the law, are legal, went through the courts and were originally on record. Whether those records still exist is another subject entirely.
"The Catholic Church, Church of England and the Salvation Army ran, “mother and baby homes” and UK adoption agencies".
Mother and Baby Homes, yes, no argument with that detail at all. Adoption agencies though, well, this is a very grey area. All those institutions you mentioned were very closely connected for sure, and the agencies involved often had a religious ethos, and often they had the blessing of the church/local diocese, but to say the church ran such agencies is stretching the truth, and often it's simply wrong.