... and these gave some clues.
The frame of your bike has too much "fancy" bits for it to be a purpose built HD bike. Especially the detail around the ball-head and the subtle sopport running from the rear axle up to the sadle. Furthermore there should be mounting brackets on both sides of the handlebar to attach the rear brakes to. Yours obviously has a fixed rear hub with kick back brakes, right?
This however does probably make it an "old" bycicle, and as such very well useable as wartime transport. It is genarally very hard to actually date a bike, because the frames themselves have changed little over the years, and accesoiries are easily replaced/added over the years.Try cleaning the rear hub, this normally hides some info on the manufacturer.
Besides a New Hudson HD bike i also have a pre-war civil bike, now in use as a RAF arfield runnabout. It is actually quite special as it has a Sturmey-Archer 3 spead K-type hub, from the early thirties, cable operated with this tiny lever on the horizontal tube of the frame.
This is a rather pour picture of the New Hudson, but you can just see the connecting rod for the rear brakes below the frame. The lamp is a late twenties candle stick lamp.
Martin.