Going back to the original poem, Nerd's point was that it's the elimination of the 'Nichol' character that has made a nonsense of this verse.

When Nichol talks about the wars,
It's war than deeth to hear him;
I oft steal out, to hide my tears,
And cannot, cannot bear him;
For aye he jeybes, and cracks his jwokes,
and bids me nit forsake him;
A brigadier, or grandidier,
He says, they're sure to meake him.

However, unlike Nerd, I think that it's Nichol, not Jemmy, who is cracking the jokes, bidding her not forsake Jemmy, and assuring her that Jemmy will no doubt receive a promotion.

The tune does indeed have several phrases reminiscent of 'Sweet Thames'.