Samirich- He did not.
The Christmas song seems to be a more recent use of the tune "Well-a-Day."

Tune "Well-A-Day," used for "Essex's Last Good Night," given on page 176 of Chappell, for a song called "The Lamentation of A Sinner," a song sung by a widow in the play, "A World's Folly." This was for the father.

The son, Robert Lord Devereux, topped in 1601, would have known a version or three of the song "Well-A-Day," but the lament was composed later and published two years later in 1603, printed for Margret Alide.
First verse:

Lament, lament, for he is dead
Who serv'd his prince most faithfully;
Lament, each subject, and the head
Of this our realm of Brittany.
Our Queen has lost a soldier true;
Her subjects lost a noble friend:
Oft for his queen his sword he drew,
And for her subjects blood did spend.
etc.

The tune dates back to the early 1500s at least; except for the oft-used lines around 'Well-A-Day,' the early versions seem to be lost.

PM Malcolm Douglas; he can tell you more.