I have examined the census returns for Yorkshire and had little difficulty finding Harry Cocker, 1885, in the 1901 Census for Halifax. He is living with his aunt, Mary H Bailey, at 3 Cotton Street, Halifax, and is working as a Draper's Shopkeeper's apprentice. His Aunt is described as single rather than a widower which tends to suggest that his mothers maiden name was Bailey rather than the proposed Iredale.
There is no sign of John William Cocker in Yorkshire. There is only one Mary Ann Cocker and she is married to Maurice Cocker in the 1911 census for Sheffield. This family appears to be too small as she has only had 5 children.
The family, apart from Harry, appears to have vanished. In these circumstances I suggest you start looking at passenger lists starting with the main emmigration routes to USA and Australia. There are a number of online sources you could consult. Try
Just google "passenger lists" for the many other possible sites.