I saw that programme too -- he's very good, isn't he? Made me come over all hunter-gatherery.
I'm not sure he's completely right though -- perhaps a telly simplification? It is certainly illegal to hunt birds with a bow (Section 5 of Wildlife and Countryside Act 1981, and perhaps earlier Acts). Also deer -- these (even muntjacs!) must be killed with a rifle above a minimum power, which rules out bows as well as smaller rifles, shotguns, catapults etc (don't know which legislation, but there are various Deer Acts).
I'm not sure about other animals such as rabbits though. About 25 years ago I had a friend who was a competition-standard archer, and he regularly shot bunnies with rubber-tipped arrows (target ones apparently go through without killing them quickly enough, but the solid rubber hits them so hard they die instantly). Don't know if it was legal. I've also seen people with similar rubber-tipped quarrels for a crossbow.
Of course if you did it round here you'd have your fingers cut off and be put in the stocks for a week...