The item you have found was the Leach catapult manufactured by Gammages' (the department store) sports department in some numbers and was used in action in 1915. It was designed by Claude Pemberton Leach. It launched what was called a 'cricket ball grenade'. Range was about 150 yards. Opinions of its effectiveness vary, it was certainly light and could be carried forward and used to consolidate positions in captured trenches by breaking up counter attacks. It was also useful at night as there was no clue as to where the fire was coming from (no bang, no flash, no smoke). But in some parts of the front no mans land was sufficiently wide that it couldn't reach the opposing trenches. They were used succesfully at Neuve Chappelle by the Royal Irish Rifles, the Linconshire Regt and the Royal Berkshires all who seem to have sworn by it (other units seem to have sworn at it). However the elastic perished after a time. Vulcanisation was tried but this reduces the 'twang' factor (trust me I used to work for Dunlop).
Is an illustration from one of my (unpublished) books showing how a Leach was used propped up against a sanbagged parapet (although a metal stand was available).
The Australians (as with many things) claim to have invented this first at Gallipoli, using such a weapon at Quinns post in October 1915 but this seems to have been a case of parallel evolution. The Australian weapon had a range of 120 yards. At the same time the engineers of the Royal Navy Division’s 2nd company (also in Gallipoli) were building Leach catapult look alikes and these were used by magnificently named Royal Marine Cyclist Company.Looking at the history leach got their first.
The French also used rubber catapults but theses had a much shorter range and were not very succesful. .
I have found this photo of a similar device used on the italian front
The caption says: "Istruzione relativa al funzionamento della Fionda lanciabombe sistema Cerulli" = "Instruction for use of the sling bomb-trowher model Cerulli" (I guess Cerulli was the man who patented it).
The book says this type of device was rare but was used on all fronts, they could launch handgranedes to a range between 80 and 100 meters.