It's taken me a while but I will post it anyway.
Tree on Ancestry.
William Edward Walker
19001981
BIRTH 18/03/1900 West Kirby, Merseyside
DEATH 16 JULY 1981 Sheldon, Birmingham
Date correct on 1939 register in Birmingham but no birth record in West Kirby, 4 others elsewhere 2 in London & 1 in Kent, however below looks the most likely in my opinion.
WALKER, WILLIAM EDWARD -
GRO Reference: 1900 J Quarter in ASTON Volume 06D Page 248
As the mothers maiden name is not listed I think he may be the illegitimate son of Mary below.
1901 106 Tilton St. Deritend, Aston, Birmingham (Mary is Mary A on 1881 and Mary Alice on birth record)
William Walker 49 Cabinet Maker born Market Drayton, Salop
Emily Walker 46 Stafford
Mary C Walker 21 Daughter born Market Drayton, Salop
William Walker 1 Noted as son
William & Emily both died before 1911.
No trace of Mary or William.
Pure speculation of course but his birth or marriage record will confirm as mentioned by others.
Colin