@ConquestEmpireHungerPlague - If your missing baby's mother was in service, then I'd definitely suggest looking more closely at her employer. Perhaps the baby was biologically "the man of the house"'s child and the family took them in? Or you might try to find out if there were any foster parents in the town/village the mother lived in at the time - because you might find the baby there, with them.
My great-grandmother was in service, in the next city over (having been born and raised in a small, rural setting), when suddenly she reappears in her family home, pregnant. She was the youngest daughter, very spoiled (her mother died when she was 7 and her oldest sister raised her and the two boys younger than her, whilst the other siblings either went into service or worked the land alongside their father) and I think undoubtedly out of her depth completely. She has the baby at home, registers the birth a month later, but the baby is placed with a couple in the rural setting who were unofficial foster parents. I don't know if my great-grandmother had any intentions of reclaiming her daughter or not, but the foster parents upped and left the rural setting and... all contact was lost. The baby girl was essentially stolen - but in those days, what could my great-grandmother have done, without drawing attention to the fact that she'd had a baby out of wedlock and disgracing not only herself, but her family, too (hate this fact, but it's sadly true - even in '51 when my mother was born "on the wrong side of the blankets", she was bullied and teased in the same rural setting until she learned to fight back). My great-grandmother goes back into service, working for another couple in a different place - this time an upmarket spa town in Wales.
Roll forwards a couple of years, and she's marrying a man whom she barely knows - my great-grandfather. No clue as to how they met - she was a house maid for a well-to-do family (in a beautiful house, which one day when this pandemic is under control, I intend on marvelling at the outside of in person!), and he was a railway plate layer. However, my great-grandmother was pregnant, so perhaps the haste can be explained. DNA shows that I am connected, albeit distantly so not a direct line, to the well-to-do family my great-grandmother worked for. Yep, the baby that caused her marriage to my great-grandfather... was not his!!! And he knew it. Or suspected it enough to insist that the baby was "given" to her oldest sister (who was already raising two other sisters illegitimate children - the poor woman had two teenagers on her hands and must have been thinking "freedom is in sight...", when she had a legitimate baby thrust into her sole care because he wasn't biologically the child of the man whose name is on his birth certificate!). Within a year, my great-grandmother had another son, whose parentage my great-grandfather must have trusted completely because they kept him. Baby, after baby, after baby came along until my great-grandmother died, giving birth to twins (who also died), aged 43 years old. Living in what was then the "slum" area of Swansea, married to a man who according to my grandfather (the middle child of 9) lived up to the cliches of a miner. He drank, he cheated on her, he beat her and the boys until they were old enough to defend themselves. My grandfather rarely spoke about his parents, but he did tell me about his mother's death and a few other bits when I was growing up (and undoubtedly too young to hear them, but he probably thought I'd forget what he said, because I was too young - so a safe outlet for him). His then-oldest sister was 17 when my great-grandmother died, and she took on the care of a 5 and a 3 year old. Just like her aunt had taken on the care of her mother and uncles.
One of the fostered out daughter's sons came knocking on my grandmother's door shortly after the headstone had gone up on my grandfather's grave (which he shares with their son, who died at 6 days old). The inscription sort of implies that my grandmother was a childless widow and, at the time - never having heard anything about this woman - everyone thought this man was simply after my grandmother's money. Which wasn't entirely baseless as he conned thousands out of my grandfather's youngest sister within the next two years... 
It was the DNA and the fact that one of the man's nephews, the missing daughter's grandson, turned up as a cousin, however, that actually made me sit back and have a "woah!" moment.
I hate to think it, but I suspect my great-grandmother may have been raped, and this missing daughter - the only illegitimate child in that immediate family to have been placed outside of the family - was the result. Something awful must have happened for my great-grandmother's sister not to have taken her in. And then my great-grandmother moved to another town, another place in service... and again, gets pregnant, marries someone who wasn't the baby's father - and has no recourse once he realised this fact. She couldn't afford to divorce. Her life was horrible once he changed jobs and moved the family miles away from everyone she knew, her family, the two older sisters who would have been her support bubble (to borrow new terminology). Their children all hated him. The boys joined the Army as soon as they were able to (my grandfather was 14 when he signed up) and the oldest daughter married at 16 to - as she once told me - "get away from [her father] and his fists".
But yes; informal adoptions happened. In my family's case, the missing daughter was loved, well-cared for and happy - which is the main thing. She also never knew that she'd been adopted, according to her grandson, it was only the DNA testing which has joined her children (apart from the conman - but he was already estranged from his own siblings prior to any of this, so definitely not a nice man) and descendants back into the fold of her biological mother's family. No clue (yet...) as to who might have fathered her, though. That's our next hunt, but as you'll know, it takes time, hunches kicking in, sheer blind luck sometimes for connections to be made.
Sorry @WeatherwaxOn... 