Roger Packman sent me details of his great grandfather, Henry Tillett Packman, a wealthy butcher and meat salesman (top picture), who in 1901 lived at number 9 Tavistock Square with his family and servants. Henry and his wife Ann had five children - Annie Wensley, b. 1881; William Henry, b. 1883 (Roger's grandfather); Nellie May, b. 1884; Gladys Mary, b. 1886; and Winifred Dorothy, b. 1890. At the time of the 1891 census, the family lived in Brixton, South London, but moved to Bloomsbury sometime during the next decade.
Henry Tillett apparently owned the Tavistock Theatre, for which I can find no information although Roger says that another family member has theatre programmes printed with family names. These might throw light on the whereabouts of the theatre.
However, a very interesting theatre was based in Tavistock House at number 1 Tavistock Square, several decades before the Packman's moved there. This was the house where Charles Dickens lived from 1851 to 1860 and where he wrote Bleak House, Hard Times, Little Dorrit and A Tale of Two Cities. Dickens converted the house's large schoolroom into what was billed "The smallest theatre in the world". More information about this theatre can be found at: http://www.victorianweb.org/authors/dickens/gallery/30.html
Roger also sent two photographs of his grandfather, William Henry Packman, in two football teams of 1905/6 (in which he is seated first right, front row) and 1910 (standing third right, back row).