If we assume that William was truthful about his father's details when he married Doreen, then I think that Alfred Shuter, carpenter and disappearing husband of Fanny looks like the best candidate.
He is probably the Alfred Frederick Shuter registered in 1860, mother's maiden name Davis.
There is a marriage in 1842 in West Bromwich district of William Shuter; Ann Davis appearing on the same page as per the FreeBMD index. There are some trees on Ancestry which agree that William Shuter married Ann Davis, although no one seems to have a marriage register copy. They also agree that Alfred Frederick is their son, but have Alfred's wife's name as Fanny Daisey. There is no evidence for this actually being her name.
Although Alfred does not always use his middle name (eg when he married), his death in 1937 is registered with both names.
I have not been able to find Alfred in 1901. It appears likely that he had already left his wife and family by then. He may perhaps be with a woman named Lettie Hill.
Alfred and Lettie Shuter baptised a daughter named Lily at St James, Handsworth on 19 February 1908. They live at an address in Alfred Street. Alfred is a carpenter.
The birth of Lily Shuter, Mmn Hill was registered in March Q 1908 in West Bromwich district.
Lily is not with her parents in 1911, but as she grew up and married a William Cook in 1932, she must be somewhere. (She shows up in Smethwick 1939 living on the same street as Lettie (whose name appears as Shirter. Lettie died in 1968)
The only other Shuter birth registered between 1890 and 1910 with Mmn Hill is a Cornelius in June Q 1895 in Birmingham. I can't see any likely Shuter - Hill marriage.
No sign of Cornelius anywhere after 1895 (not even in New Zealand!), until a Cornelius Shuter death is registered in Worthing district in 1968. He is aged 72.
The National Probate Calender for 1969:
SHUTER Cornelius otherwise William of 35 North La East Preston Littlehampton Sx died 14 February 1968 Administration London