Wow, I am surprised it was not pedal powered! Most certainly a novel use of the Sturmey Archer as a gear box for a lathe.
The later lathes shown on lathesdotco remind me of the drives for sheep-shearing cutters or forge blowers.
I agree with Thor – if there is sufficient distance between centres, one might even be able to cut a new spindle!
A simple whitworth nut, with a tag welded on, could suffice as a very crude drive dog.
Likewise, a large nut, welded to a disc, might well suffice as a faceplate or a backplate for any suitable small chuck.