I’ve stumbled across the only baptism I can find for the Cook family.
Joseph Cook Born21 Feb 1875 Baptised14 Mar 1875 Abode- Phillip St. Joseph, (stamper)Cornelia
Joseph Cooke Birth abt 1875
Death - 4/1877 Birmingham Warwickshire
This puts a question mark for the birth of Henry Cook Q4/1874 Birmingham – Vol 6d page 172
The only one I can see earlier was born Q4/1871 or was he William Henry Cook born 4/1873 Birmingham
Colin
PS. Henry G Cook is also on the 1881 census:
Henry G. Cook
Mother: Eliza Cook
Father: Stephen Hy. Cook