According to a note included in the box, this was made by Charles Holtzapffel. Reference to Holtzapffel's work on display at the Science Museum dates the note to after 1928.
The pattern on the base is reproduced in Engleheart (Eccentric Turning, 1st Edition, 1852, Fig 36). If the note is correct, then Engleheart was merely publishing an existing Holtzapffel design (Charles Holtzapffel died in 1847).
As the author of the note shows an incomplete knowledge of the Holtzapffel family, it seems likely that there may be a connection between this box and the family or the firm, but not necessarily with Charles in particular