Hi again Jean.
I think you're right. The William Illman I found was the son of Henry and Louisa Illman. Henry Illman, a gardener, was born about 1845 in Reigate, Surrey. In 1891 the family were at the same address - 3 Belmont Grove, Chiswick. [RG 12/1034 folio 26 page 15]
Going back even further, Henry appears to have been the son of James and Ann Illman. In 1851 he was living with his parents in Reigate. James Illman, a blacksmith, was born about 1813 in Chipstead, Surrey. [HO 107/1599/1 folio 205 page 34]
So the William Illman in the picture seems to have been James's grandson.
Dave