Hi Giacomo,
The Rudge is a 1913 "TT Roadster" - the sporting man's choice. Single gear, no clutch, belt drive.
A good starting point for identifying any motor is the exhaust pipe. Notice on the Villiers is comes out horizontally at the front, half way up the cylinder. Only a two stroke has an exhaust in this position. The Rudge has a very high, left facing exhaust which is most unusual on a veteran bike.
Other give-aways on the Villiers - the muffler under the motor with double expansion chambers, outside flywheel on the left, cylinder shape and so on. The magneto appears to have two "knobs" on the alloy end plate. This was a feature of the British-made Thompson-Bennett magneto, which in the early 1920s became the first Lucas magneto. So far as I know, Thompson Bennett was one of the many brands of magneto that sprung up during the war - pre-war most magnetos were German, particularly Bosch. In fact the magneto on the Rudge is a German Ruthardt (the design of which was "stolen" by the British firm CAV, often seen on early 1920s Nortons). In 1912 Rudge used the German Eisemann magneto.
Anyway, I understand why dating the is so important. From the Rudge, we know that the photo cannot be earlier than 1913. Other than my notion that a Thompson Bennett magneto would indicate 1915-on, there's no reason why the photo couldn't be 1913-14. Does anyone know when Thomspson Bennett built their first magneto? Perhaps more info will come from the registration numbers.
Leon