According to another website and a thread on here, the oldest song which uses that tune (County Down/Dives and Lazarus) is Gilderoy, which goes back to Allan Ramsay in the early 1700s.
I know of at least 5 songs to that same tune, most if which are listed on a Wikipedia page about Dives and Lazarus, and include Unquiet Grave. Not on that list is Van Diemen's Land (Gallant Poachers).
The first version of Unquiet Grave that I heard was Joan Baez's recording, but it uses a totally different tune.