Hi Ian,
Unfortunately I'm familiar with both Bowden and Burman gearboxes, and these boxes are not from either of those makers...
I'm not sure if Douglas owners have noticed, but the usual Douglas layout of primary drive on the opposite side to the final drive (i.e. a "cross-over" gearbox) is not the usual way of doing it. Of reasonable sized manufacturers, Sunbeam and Douglas were probably the two largest to use cross-over boxes. (Triumph dabbled in the layout in the mid 1920s.) Sunbeam and Douglas weren't big, but they did build some very fast bikes - interesting that both Sunbeam and Douglas "specials" tended to use gearboxes with final drive on the same side as the primary. With cross over drive, the primary drive pulls forwards on one side, and the final drive pulls back on the other, both twisting the gearbox in the same direction around its mount. The big makers of gearboxes in the early days (Sturmey Archer, Burman, Albion, Jardine, Moss, ...) rarely used the cross-over format.
The two boxes here have some amazing similarities with early Douglas boxes: cross over drive, vertically split cases, two speed, vertical selector shaft on the "split line" of the case, clutch in the final drive pulley, kick starter shaft carried on an extension of the cases, ... If it weren't for these similarities I suspect the Administrators would have booted us into "The Kingswood Pub" a few posts back!
Cheers
Leon