He had a younger brother called George who joined the Beds Reg
Picking up from the army record, the brother George Alcock
lived in Bedfordshire. There was also a sister Mary (aka Polly). Both children were in the care of the Stepney Poor Law Union. They were boarded out by the union from 16 May 1884 onwards to three successive foster carers in Bedfordshire. They were then sent into service, in 1891 and 1893 respectively (Mary to Grantham, Lincolnshire).
(source: LMA, STBG/L/146, via Ancestry > London, England, Poor Law and Board of Guardian Records, 1430-1930 > Tower Hamlets > Stepney > Register of Children > Register Of Children Boarded Out, 1876-1889, pp. 81-2, and 1882-1908, pp. 19-20)
I did not find William in the above poor law records. It is worth noting that when he attested for the Lincolnshire Regiment in 1887 he was in
Dublin.
George and Mary are found as follows:
1891 censusRG12/1260 folio 106 page 43
Green End, Maulden, Bedfordshire
Eliza Hickson, Head, Widow, 60, born Maulden Beds
George Allcock, Boarder, 14, born London
Polly Allcock, Boarder, 10, born LondonAlfred Robinson, Boarder, 13, born London
George is found in Luton in 1901 (again with Alfred Robinson)
1901 censusRG13/1518 folio 117 page 30
19 Boyle Street, Luton
Thomas Osborne, Head, married, 38, Straw Plait Dyer, born Beds Totternhoe
Jane Osborne, Wife, married, 31, born Beds Eaton Bray
[3 children, born Luton]
Alfred Robinson, Boarder, single, 23, Plait Dyer(?), born London
George Alcock, Boarder, single, 24, Straw Plait Dyer, born London
Perhaps you can take them further from there?
As there is no father named on William's birth certificate, it's possible that the father's name on the baptism record was an invention, for respectability's sake.