Hope my but-in does not hinder !
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My father's birth was registered in WW1 era (1918).
Looking at his copy birth certificate, I see that his surname was not recorded.... no column for it !
Column 4 is for " Name and Surname of Father"
Column 5 is for " Name and Maiden Surname of Mother"
For the child, Column 2 is for " Name(if any) ".
My father's first name James, is recorded here with no surname.
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Looking at my birth cert, 1941, I see that ditto applies to me. Also to my grandfather's 1875.
( I cover a range of years in case the info recorded on registration, changed over the years ). I have a hunch that today children are registered with name and surname ??
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But as far as indexing of registrations goes, he would have to have been indexed under some surname.. Dixon in his case.
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To find the birth registration of a child whose surname is unknown would be very difficult, but maybe not impossible !
From the quarter ending Sept 1911, GRO birth indexes included both the child's father's surname and the mother's maiden surname.
Where the mother of the child was not married to the child's father, the birth registration index would include the mother's maiden name twice.
So ( assuming you have dtr's first name and an idea of where and when born/registered) get into those indexes !
Would the addresses of your WW1 relative and the girlfriend afford any additional clues as to the identity of the child ?
Michael Dixon
Newcastle