Presumably the bulb has a spring inside so that it acts as a bellows? In that case, would it have had a rubber bladder inside (now defunct) that the cloth covered? I doubt the wartime story, and it looks more like a bicycle horn.
The only bulb horn that I've seen documented in connection with WD motorcycles is the Lucas 60.V specified for Egypt and Malta. This one looks to have the brand name 'Omen'. The registered design number could be followed up and may well explain what and why.
It may well be wartime, but no reason to suppose that it's WD.