Julie he may still of been buried at Eldon just that he doesn't have a headstone or its been removed.
Do you mean Rosemount Road Cemetery? I don't think that was open in 1916.
The burial register for Eldon for 1916 is held at Durham Archives, you could contact them & ask them to check it out, tho not having his death date they will charge a search fee. If you have the date they may check it out for free.
Good luck
Jay