Well in the case of a neighbour, they ended up at the hospital where they took the body. The police then investigated it and decided on no post mortem and the body was released. However before that occurred the next of kin were allowed to have the possissions that were not actually on the body, like rings, as they often have to be removed afterwards unless the relatives decide to let them go with the body.
So the ananswer is, it depends in many ways, certainly when my mother died of a sudden stroke, I collected most of her belongings at the hospital, and the rings from the undertaker, as in that case as with all sudden deaths the police were called but took no action other than to come to the house and tell the next of kin.
Brian