As I see it, so long as the owner holds a gun license, and to prolong the animals life by waiting for a slaugherman, or vet to arrive, would cause the animal suffering, then it is permitted for him to shoot it. If he was just doing it to save paying out for someone else to do it, thats a different matter, but it would be proving it.
I know when my friend found her horse down in the stable with a broken leg, she was distraught waiting for the vet to arrive (although I am sure he came as fast as he could), In circumstances like that, I think I would be grateful for someone to put my horse out of its suffering quickly. That is, so long as they shoot to kill first time.
As to burying, I think you may need the permission of the local council even if its your own land. And it must not be near any drains or water courses. Having said that, numerous people near us, use their JCB to dig a hole and bury a dead horse, and I am sure they have said nothing to anyone about whether they can or not.