They might not have been saying 'thee' in
London when the Davy Crockett was plying the seas, but they
sure as heck were in Manchester--- and, one supposes, over in
the 'Pool.

Theein' an' thouin' has never (in my lifetime) been associated with Liverpool, probably because of the Irish influence. The second person singular form seems to have changed about the end of the 17th century, and only held on in conservative communities- agricultural areas or areas where the mills and mines developed out of an existing tradition, like Lancashire and Yorkshire.

Quakers of course stuck to the thee form for many years, having adopted it as a classless address (and suffered for doing so), only to see the rest of the language jump the other way. And it only dropped out of poetry when poetry stopped being prissy.

So I suspect that the singer through the porthole sang either "my darling when I think on ye", which would both be standard usage and rhyme, or "..thee" as a poetic gesture.

I suppose Americans would wonder about the function of another Lancashire character of my grandparents' generation. His function was exactly what the title said. The knocker-up went round in the early morning, knocking people up.