Just to add an update, during the summer, I came up with a new theory. I shared this with the family and a professional family historian hired by the family has agreed with it. There are records going along with this that I won't mention to make it succinct for anyone interested.
My theory is that May's parents are (at least on paper) Thomas John Collins (this could need looking into) and Florence Eliza Steer. There is a birth record for this as Florence May Collins. This is because my family have strong DNA relationships to the Steer family and also the descendants of May Dorothy Collins as well as the shared matches aligning. From these DNA matches, we can identify records.
Now, the part about Albert Collins (the reputed father) going to fight in the Boer War and coming back to find a baby that wasn't his is possible - just that Albert was actually Thomas John Collins.
Thomas is in South Africa at the time of the 1901 Census, so the household back in London includes
Florence Eliza Collins and daughters Maud and May. Descendants of Maud do match descendants of May highly, but also to my family. However, by 1904, they are in the workhouse and subsequently the census record near to the start of the thread is correct.
My theory is that Florence Eliza ran off to Kent, changed her name to Mary and had a child with Joseph Yielding (who died before the birth of the child) and then 5 more children with Thomas Smith (my family). This is certainly what the DNA evidence suggests.
Thomas John Collins did return to London and get in contact with his daughter Maud Collins, but not May. He did go on to build a family with a Frances May Jessup. Hopefully, a descendant of them will read this and say that they have done a DNA test. If they do not match the family of May Collins, then we know that Thomas John Collins was not May's father; if they do, then it confirms the father.
