I don't think it's expressly 'legal', as opposed to not being prohibited by law, with all the usual caveats as to intent to cause injury or damage, etc. I do, however, believe it's illegal to use pebbles as ammunition, as that's expressly covered under 'stoning an animal to death' in the Wild Mammals Protection Act 1966.
What's really daft though, is that you can walk around with a handful of ball bearings and a Black Widow while completely within the law, but if you were to be caught anywhere having accidentally forgotten to remove your knife from your pocket you're immediately guilty of a strict liability offence . . . . .
English law makes no sense and has been built up over the years on knee jerk reactions by politicians. And there are far too many laws. So for example all the knife laws came in as a result of kids running around with knifes and stabbing each other. There is a perfectly good laws Murder, Manslaughter, Gross Bodily Harm and Assault which cover both intent as well actually committing. These would actually cover pretty much all of the offences intended and or committed, and proper policing and education would cover the rest. And I deliberately use policing and education in the same sentence, as a large part of policing should be about nipping crime in the bud.
But there is a media outcry etc etc etc and rather than actually adressing the real problem of kids growing up without any life prospects other than joining a gang, it is so much easier to satisfy voters but passing a law that knife blade which doesn't lock and is less than 3" long is perfectly safe and legal, yet a 1" blade which locks is deadly. Such laws have zero effect on the perpetrators, yet have major consequences for everybody else. And it allows the Police yet more powers to arrest and prosecute so that the crime solved numbers look a hell of a lot better.
Rather than judging police on numbers of crime solved, prosecuted and won through the courts, the real measure should be how few crimes actually need investigating. A really good police force should have plenty of time on its hands.
As regards an object used in a threatening behaviour, to be honest there are plenty of tools that can be used in addition to your hands and teeth to threatening or cause harm etc. You could ban cotton wool because you could stuff that up somebodies nose and mouth, and as has been shown on numerous occassions a car, truck or aeroplane makes a lethal weapon.
Coming back to hunting tools - of course a bow or catapult is perfectly capable of humanely killing animals and birds, if used by hunters skilled in their use. A catapult has probably similar energy levels to an air rifle, but takes longer to master the skills to use it, and because the ball doesn't usually penetrate it's really only useful on smaller animals / birds. For big animals, you are talking a between the eyes hit and stunning the animal. And a decent bow, whether it be an old English Long Bow or a modern compound bow is perfectly capable of killing a large animal at considerable range, and if you put the arrow in the kill zone, severing the major arteries it will be dead very quickly. Put a 300 Ultra huge velocity magnum with a super deadly bullet through the guts of any deer - it will run off and die a slow death.
One day there does need to be a complete revision of our somewhat archaic hunting, firearms and licencing legislation as it really is not fit for purpose. In most other countries the focus is the on the individual and whether or not they have the right aptitude or skills, with need to acquire a hunting licence which then allows you to possess and use the tools as appropriate.
Personally I think Bows, or even cross bows would be a very useful tool in armoury for deer management, especially in the more densely populated areas as it is silent, and whilst deadly out 50 / 75 yards, an arrow runs out of steam pretty quickly thereafter. But it, as always, it is down to the individual behind the weapon to ensure a clean humane kill - no different in using a bow at 25 yards or 75 yards, to a rifle being used at 50 or 300 plus yards.