Is it permissible or forbidden to transport the bodies of the deceased from the hospital to the church and then to the cemetery, without touching the deceased in any way?

It is not permissible to transport deceased disbelievers to a church to assist them in sin, for the church is a house of falsehood and polytheism against Allah. As for carrying and transporting a disbeliever to the cemetery, the Hanafis permitted hiring someone for this, because it is a carcass whose harm should be removed from people. However, they do not permit transporting it from one town to another. The Hanbalis, on the other hand, deemed it forbidden to carry the funeral of a disbeliever, or to shroud them, or to follow their funeral, or to pray over them, due to His saying, the Most High: "You do not befriend a people with whom Allah has become angry."