You grandfather was obviously a man of taste and discernment owning not one but two Villiers powered beasties. Much smoother and nicer than those coarse 4-stroke thingies.
Unfortunately identifying early Villiers powered machinery is very difficult as so many makes used them.
I have in front of me the 1920 Olympia Motorcycle Show catalogue opened at a full page Villiers ad. In it Villiers proudly announce "Fitted to Ixion, Sparkbrook, Hobart, Diamond, Victoria, Challenge, Bowns, New Ryder, Invicta, Coventry Eagle, Chater Lea, Wolf, P.V. Revere, Excelsior, Campion, O.K. Two-Stroke, Lloyds, etc, etc." So you see identifying an early 20's Villiers powered bike is not simple. I suspect the bike in photo 2 could be a Hobart but I'm far from positive.
Cheers,