The information on certificates is only as good as that supplied by the people involved The civil registration system in England and Wales is 'informant driven' that is the registrar/priest can only put what he is told. There is a penalty of perjury if the information is willfully false,
From the 1836 Act.
XLI. And be it enacted, That every Person who shall wilfully make or cause to be made, for the Purpose of being inserted in any Register of Birth, Death, or Marriage, any false Statement touching any of the Particulars herein required to be known and registered, shall be subject to the same Pains and Penalties as if he were guilty of Perjury.
Between 1837 and 1875 if the mother informed a registrar of an illegitimate child's birth and also stated the father's name, the registrar could record him as the father, otherwise the space for the father's name and occupation will be blank.
This applied until The Registration Act of 1875 which stated:
"The putative father of an illegitimate child cannot be required as father to give information respecting the birth. The name, surname and occupation of the putative father of an illegitimate child must not be entered except at the joint request of the father and mother; in which case both the father and mother must sign the entry as informants"
Stan