"Oh what a tangled web we weave" ... we poor amateur genealogists don't stand a chance!
Which surname do this man's descendants search for?
His WW1 service Records (WO 363) are filed under MARKHAM.

He believed he was called William Charles MARKHAM, born Berkhamsted.
In fact his name was William Charles WALKER, born 3 June 1899, Winkwell, Hemel Hempstead.
His mother, EMILY WALKER, was an unmarried Domestic Servant in 1899 - and in 1916 did not want her son to know he was illegitimate.

Working as a Surveyor's Clerk, studying for his Architect articles, on 5 August 1914 15-year old William Charles enlisted with the Hampshire Cyclists Regiment, declaring age 17y 1m. He served overseas until his mother wrote to the Army (with his Birth Certificate - cf page 35101) and he was recalled in 1916 (by which time he was a Lance Corporal).

When he was of age he wanted to rejoin his old unit, but had had an accident which affected his eyesight. He was appointed to the Labour Corps where he served until discharge in 1919.

Home address: Boscombe, Hants (1916) & Eastergate, Sussex (1919)
"Father": J W MARKHAM
Mother: EMILY nee WALKER. Probably married MARKHAM, but her surname in 1916 was MARTIN.
Regt Nos: 588 Hants Cyclists; 14701 Army Cyclists; 475031 Labour

Anc****y, MAR, pages 35054-35117