Brummie 60,
Sorry can't tell you where the name Cooksey came from, but I can tell you that Cooksey Road, Small Heath was adopted by the Corporation in 1878, and first appeared in the Kelly's Directory for that year. Part of it still exists, running about 500 yards west of Golden Hillock Road, but it once ran a long way west to meet Bolton Road and Bordesley Park Road. I have found two references to Cooksey Place, in Bordesley Park Road (probably just a group of buildings) in the 1880s, and to Cooksey Avenue, a short road off Cooksey Road, first appearing in the directories I have been able to check on in 1921, which disappeared in the 1970s, I believe.
The Cooksey Lane in Kingstanding was in the 1933 directory but it masy be a few years earlier - about 1929, I think.
I can also confirm that there is no area of London known as Cooksey, nor are there are no roads, streets, lanes, avenues etc bearing the name of Cooksey in Greater London today, or in 1969.
Not much help, but at least it confines the area of our ignorance.
Peter Walker