TBH, although I've never hunted with a spear, I think it can't be much different to using a bow and arrow, and hence quite difficult to make a quick kill.
It is very easy to make a quick kill if you are using the correct equipment, way back in history neolithic man discovered that by using a broad tip and aiming for the heart they didn't have to follow a dying animal for ages to get their dinner, a good clean shot especially with modern hunting tips will bring them down very quick
The Wild Mammals protection Act 1996 makes it an offence amongst several other things to impale ,stab,beat,crush,or stone a wild animal. Whether using a catapault constitutes some sort of stoning might be debatable especially if you were using ball bearings as ammo rather than pebbles . The Act does state though that is the intention to cause suffering through these acts that creates the offence .
In theory a catapault used with the intention of making a clean kill creates no offence as long as the quarry is a legal one and providing the catapaultist (possibly a made up word ) is authorised to be hunting on the land . Authorisation comes from the owner or controller of the Land . In the case of "Public " land it will almost certainly be overseen by a local council or nature management group of some kind and permission must be sought from them .
A classic case of this came a few years ago when an enterprising chap was catching London street pigeons to sell to restaurants . Although they were a pest species and were caught in a public place he had no authorisation from the local council hence was prosecuted .
Although the 1996 Act mentions impaling and spearing as offences and the 1981 Wildlife and countryside Act specifically mention the use of bows I'm pretty sure the ban on hunting with bows dates back to the late50s or early 60s and is connected to the deer protection legislation that came in about that time .