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When it came time for Sylvia to get the pension she had a problem because she didn't have a birth certificate. I don't know how she overcame that.

James Bright aka James Wylie was my grandmother's(Frances) oldest brother. In the 1911 census James Wylie was the head of the house and Sarah Martin was his mother. Florence (died 6/03/1913 same day as the Wylie/Martin Collins marriage) and Lily were his sisters.

The memory this child had could have been from family conversations and not actual recollection of an event

You have said that family stories claim Sylvia living with her Mother in London.
Louie's address on wedding certificate is Southend, not London.
Do you mean visiting her after the marriage?
Does not make sense.
Why would they marry in 1913, then emigrate only 2 years later without her - unless she has died?
Could Sylvia have been born in Ireland?
If Louie had died then they could take Sylvia even if she is not Robert's/James's daughter?
When you say about not finding her birth, have you applied for any birth certificates?
Robert/James seems to be the one changing names, as family keep the same ones until after the 1901 census? So, do not believe anything he says?
Any family story as to why Louie left behind?
Too many questions, sorry.

Burial Register summary
last name Martin first names Louisa Elizabeth Burial date 10 Sep 1913 Authority London Borough of Islington Location Islington Cemetery (Islington)

(possibly the 1913 Death, from below link)

https://www.deceasedonline.com/servlet/GSDOSearch

Chris :)

Family story has it that Louie/se was not looking after Sylvia to grandmother Sarah's standards so the child was taken to live at Harberton Road. Good point about address on marriage certificate. The whole things smacks of kidnap to me. Robert/James may, according to yet another family rumour have had gambling debts. You would think a silversmith would have enough money to pay his own passage to wherever but his brother John footed the bill. I haven't found any definite births to apply for a cert. yet.

In 1916 James Arthur Bright enlisted in the NZ army - embarked that year. In December 1917 he was in Chelsea Hospital (Pay book says he received 3s 6d on 4/12/1917. Pay book was signed off 1/1/1918. I have a letter from him to my grandmother dated 6/6/1918 mentioning that he was still trying to shake off trench fever. I can't find any details of his postings in NZ Army records. Would there be any records from Chelsea Hospital I could access?