"Why does the hospital think it is in charge of the body?"
Technically, in law, the body of a deceased person is the responsible of the hospital.
"Why are nurses going to police this? It is not within a nurse's remit to do this."
These will be highly trained and highly specialist transplant team nurses. In my experience seeing it from the doctors' side, they are wonderfully kind, sensitive, gentle people in such a time of grief.
These situations, especially as they are so often deaths caused by major, sudden, catastrophic events, are incredibly fraught, but it may help relatives not to feel that they are having to make a decision to (as someone said their DM said, a bit melodramatically I felt as someone who really has had their "baby cut up" in a post mortem) have their "baby being cut up after death".
It may help relatives see this as the patient's last request and take away some of their own (however we might see it as misplaced) guilt as having seemingly brutal surgery performed on a loved one's dead body.
As someone who has referred people as potential donors, I have sometimes received the letter from the Transplant team, which the relatives will as well, describing, without any really identifying details, the recipients and how they are doing.
I've never received that letter without crying at the gift of life that those donors have given to patients and their relatives.