My mum had a stillborn boy in 1951 at home. She was 7 months pregnant and as far as I know the baby was just taken away by the GP who came to see her. I did contact GRO to see if he was on their stillbirth list, but there was no trace, despite the fact that, naturally, I knew my parents' names and our address as I was 10 years old at the time. I also had to provide the dates of my parents' deaths before they would provide a certificate of stillbirth. I am going to try again, as the letter I received back said there was no record of baby W (which is my married name) and the baby would have been baby B.
When I worked as a midwife 1970s onwards, stillborn babies were usually put in coffins with other patients who had died at the hospital at the same time. It wasn't really the thing, as it is now, for the parents to take their baby and have a proper funeral and burial/cremation.