If the very precise age of 32 yrs 8 days given by William King/Eade on his Army examination (14 April 1916) is correct then his birthdate should be 6 April 1884.
Tweaking the free 1939 Register index, I think William Eade is in the Hendon area and gives his birthdate as 6 April 1885. If at least the 6 April bit is right (whether 1884 or 1885) then that could be corroborated with any birth certificate.
In turn, with luck, the birth certificate may give address (or other) details which link his mother Emma King to one of the families identified in the censuses.
Not that this helps with his birth
London Electoral Register in Hendon
1920/21/22/125/1939
31 Oak Grove
Wm Eade
Lily Ellen Eade
I think they married in Cranbrook Sep 1919 Lily E Turner
his death & probate
William Eade age 57 (
1886) Jun 1943 Hendon 3a 532
William Eade of 31 Oak Grove Cricklewood London NW2
died 3 May 1943 at Handley Page Factory Claremont Road Cricklewood
Administration London 2 July to widow
Lily Ellen EadeLily Ellen Eade of 31 Oak Grove Cricklewood Middlesex widow died 18 Dec 1947
Administration London 31 Jan 1948 to
M Sleet married woman (possible daughter bn 1923 Hendon)
Forgot to add 1921
98 Fernhead Road
Emma Eade there are others at that address but not with same surname
Kennedy, Palmer, Reed