It may interest some of you to see the three different "Laments" on broadsides in the Bodleian Library. Although, unfortunately, they give no clues to what the tune might have been.

LAMENT 1

LAMENT 2

LAMENT 3

There were also two tunes .......
Franklin is fled away
and
Franklin is dead and gone ......
in use in the middle of the seventeenth century - which may or may not have had a bearing on the later song/tune.