If this is right i.e. Lily being Annies daughter before marriage. It is most likely that she is the child of Valdimar too. Very exciting. If this is right, I might actually have some relatives alive in Grimsby! I had lost all hope of that when I was checking this all out in 2012. Even visited Grimsby and the cemetery Valdimar and Annie are buried. When I called the nursing home she died at, I was told by someone that remembered Annie that she had no immediate family. They must have just meant then that she had no child of her own. Is it common that a child that is born before marriage always carries the mothers maiden name? Even though she knows who is her father?
The story of my great grandfather was always that he had two daughters in Grimsby.. I only found Annie so we thought that was wrong.