Acts of Parliament and common law cases do one of two things; either they specify what MUST be used (as in the sections setting out specific power levels/calibres for different species of deer), or they prohibit what can be used - as in taking game with a bow. In theory, anything that is not specifically mandated or prohibited is legally allowed.
That said, courts have a wide common law discretion to reach determinations. So basically, if the act and precedent caselaw doesn't seem to cover the widget in question (eg the slingbow), the court has a discretion to apply the SPIRIT of the original intention - ie it can say that the Act was intended to prevent hunting with a bow which may be defined as a device to launch arrows to kill the game; therefore anything else that launches arrows to kill game, and which is utilised by the shooter drawing back the string/rubber/whatever - ie applying muscle energy into the device, is, effectively, a bow. Therefore using one to take game is illegal under the (I think) Wildlife and Countryside Acts.
There was - many years ago, a takedown bow sold in the US for backpackers. This was a sort of Y shape, with a long arm uppermost and a shorter arm going off at an angle and using strong slingshot-type elastic to propel the arrow. It didn't do well and dropped off the market, but it shows that the basic idea - if not the specific implementation, is not recent.