When Hampnie Hambart died, was it because his wish came true AND he succumbed to his wounds, or would he have died whether he was wounded or not because his wish came true? I'm asking because I don't recall him being that terribly wounded. It seemed like the group that captured him was just getting started.
It's hard to know for sure, but it makes sense why he would die if we accept Hampnie Hambart's own theory for what God is doing. He believes that God thinks he is granting humans their wishes. In that case, Hampnie Hambart's own wish, to die happy with his family, was granted. Until that point, he was unable to die because it would go against his wish. However, once those conditions were met, he could die just as any other human.
As for his wounds not being very severe, this is a matter of how you interpret the scene. My personal guess he was probably at the level that an ordinary human would die, but being immortal himself, he could withstand more than the average human. Another possibility is that after his conditions were met, his wish being his own death, God granted him that wish right away. What exactly the group does to him isn't depicted much so we can't know for sure whether it would kill a normal person. There may be more information in the light novels, but I haven't read them.
Of course, we don't really know what God is doing (at least not yet), so there could be a different explanation, but this makes sense given all that we know so far by episode 3 of the anime.