Sometimes you have to think laterally, you may have already checked the following but worth a mention.
Did Sarah have any siblings of childbearing age in 23/24? Sister/sister in law perhaps died in childbirth?
Have you looked for a birth using the grandmother’s maiden name.
Have you found Sarah’s birth to confirm her name was Balham?
Does the DNA confirm her maternal line?
Do you know what school Ivy attended, checked records for parents names?
Other than 1939, which doesn’t give relationships, what proof do you have that Sarah was her mother?
All Sarah's siblings were male, but I can explore whether they may have had a wife that died. There is DNA link via Sarah's grandmother, but it's sketchy because it's 3 generations up from Ivy and over via a half sibling before down several generations to the other matches, so it's possible that her mother was not Sarah but a was a blood relative of Sarah.
We have found Sarah's birth record with surname Ballham.
Thank you for the school suggestion, we do know the name of the school so I will check that out.
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.