I suspect, but am not certain, that it is a simple type of small expanding chuck ?
Drill a hole in the timber to be mounted using a forstner bit, or maybe a spade bit, would need to be a close fit. Mount the chuck in the headstock spindle and lock the spindle in position.
Mount the drilled workpiece onto the chuck and expand the grooved section by turning the wide shaft section with a spanner ( it looks like there are spanner flats on it but can't quite see) until a tight/solid fit.
Unlock headstock spindle and start up lathe slowly and standing well back :lol:
Of course I might have got that totally wrong !!!
Cheers, Paul