I think Barry is right, that Gertrude never married and that her children’s births might not have been registered.
In 1861 she was calling herself Gertrude Yates, but I couldn’t find a Lin(d)ley/Yates marriage. Also in 1871 she had reverted to using the name Linley. In the 1841 census Gertrude was on Earl St, with an Elizabeth Lindley (age 50, silver burnisher) a William Lindley (age rounded down to 20, warehouseman) and a Walter Lindley age 7. Walter is probably the William Walter Sykes Linley born 23/12/1833, baptised in 1837 together with a brother Henry Clarence Sykes Linley, born 12/12/1835 - mother Gertrude, a spinster. No father mentioned.
Gertrude’s own baptism was a late one, in 1832 (she was born 12/11/1813) A brother William (born 22/12/1818) was baptised at the same time as her, and was probably the warehouseman in 1841. Their parents were William and Elizabeth. William was a silversmith. Looks like Herbert might have given his grandfather’s details rather than admit he was illegit.