Just a little additional info to the earliest version we appear to have;
the 1769 whaler's log version is in Huntington's second volume of whaling log songs The Gam, 2014, p144. The interesting thing about this version is that as one might expect almost all of the place names are garbled in some way but still recognisable. What this tells us is that the version comes from oral tradition and must have passed through several minds and mouths before reaching the source given here. I would hazard a guess that that passage of time and geographical space, taking into account the relatively poor communications of that era would give us a space of at least 10 years from its composition. One would expect that it would have appeared in some nautical collection soon after its composition, but this has not yet surfaced.

Here's the appropriate stanza with the place names. Makes an interesting mondegreen study.

The first land we made it was called the deadman
the ramhead of Plymouth doth start London white
Sailed east beachy ship past folly and Underneys
Until we roused the Forlan light.