The 1910 Birmingham directory (and others) is available on CD from Midland Historical data at
https://www.midlandshistoricaldata.org/Booksbrowse.aspx?pg=17&ct=All&ct2=Directories#anchorM0613.
Listings for Bocock & Wilkinson from directories are below
Remember that the date of the directory is the publication date and information probably refers to a year or so earlier
Presumably the 1889 directory had an entry, but I do not have access to that one.
1890-1900 Bocock & Wilkinson, electroplate ware manufacturers 123 Gt Hampton St
(1892-1900 Joseph Wilkinson also referenced saying electroplate ware manufacturer see Bocock & Wilkinson)
1903-1908 Bocock & Wilkinson, ships lamp manufacturers, 123 Gt Hampton St
1910-1915 Bocock & Wilkinson, ships lamp manufacturers, 17-18 Mott St
(1903-1915 Joseph Wilkinson also referenced saying ships lamp manufacturer, see Bocock & Wilkinson)
1921 George Bocock & Co. ship's lamps manufacturers, 17 & 18 Mott street.
(Joseph not listed in 1921)