Now you come to mention it, I don't think we have anything saying for certain that Sarah was her mother. I have the 1921 census but Ivy is living with Sarah's mother. Our understanding is that she lived with Sarah's mother for most of her childhood
If Ivy is on the 1921 census she must have been born before 1923. Perhaps one of Sarah’s brothers got a girl in trouble and Sarah’s mother bought Ivy up. She would still be the grandmother.
Was Sarah’s mother a widow or separated at this time? When was she born? Sometimes when a young daughter got into trouble the parents passed the child off as their own, perhaps Sarah’s mother did the reverse stating the child was a granddaughter instead of her daughter.
My own great grandmother had children up to the age of 49….! Her daughter went to live with her aunt, 1911 she is down twice… as living with her mother and also at her aunt’s address. 1921 she is pencilled in as adopted daughter. Thankfully, we knew the story of why she lived her aunt and she always knew her real parents.