In an emergency e.g. if you find an animal catastrophically injured or in severe suffering, anyone can perform euthanasia for the purpose of preventing suffering, as long as they are competent, have the permission of the animal's keeper/owner and if necessary for the method of dispatch, hold the appropriate FAC and firearm etc.
If an animal is killed in more routine circumstances e.g. suffering is more chronic and less severe and the animal is not fit to be transported, the animal should be killed by someone holding a welfare of animals at time of kill (WATOK) license (and the appropriate FAC - as above). What the Food Standards Agency refer to as "slaughter of livestock with a firearm and free projectile". This animal should only enter the food chain if a veterinarian performs an antemortem inspection and a slaughterhouse is willing to accept the animal (it must arrive within two hours of slaughter). Otherwise the animal can be consumed by the keeper/owner, but not given to friends or sold for money.
An animal not entering the food chain or eaten by the family should then be disposed of after slaughter by a fallen stock collector, which it is the keeper/owner's responsibility to arrange, after killing.