Strictly speaking, Brownie McGhee was "recruited" by Fuller's manager, James Baxter (J.B.) Long before Fuller's death. But it was less than a year before Fuller's death and Fuller was showing signs of the kidney, bladder, and syphilis problems that eventually killed him. Long was certainly on the lookout for a replacement for Fuller, and chose McGhee for the role. The records from McGhee's first session were issued under his own name, but those made for Okeh after Fuller's death were issued as by Blind Boy Fuller #2. The sessions took place on May 22nd/23rd and October 22/23rd 1941 (Fuller had died on February 13th, 1941). 15 sides on 8 records were released as by Blind Boy Fuller #2 (the 16th side was by Blind Boy Fuller). Long gave McGhee Fuller's guitar after Fuller died.
It's interesting to note that on May 23rd 1941, Brownie, Jordan Webb (vocal, harmonica) and Robert Young (washboard) recorded 4 songs as by Brother George and his Sanctified Singers, a title first used by a group of Fuller's accompanists (Bull City Red, Sonny Terry, and Sonny Jones) and later on two sessions by Fuller himself (with Red and Sonny).
Details from Bruce Bastin's Red River Blues and Dixon, Goodrich, and Rye's Blues and Gospel Records 1890 - 1943.