The simple up and then down singing lines I don't think can be attributed only to the British. I have heard the Irish similarily blamed for melodies of this kind. They can be heard in some 'cowboy' songs and other simple songs with the same meter. But I presume you mean British Isles rather than British (or is this just my North American slant on the term?).
Of course, the style seems to have originated there in the British Isles, but like DNA, it is embedded also in some songs composed on the American side of the water.
I have an old Spanish song collected in New Mexico that sounds the same- the original melody forgotten.

Very little to change in your transcription. I hear:
1st verse line 2- was
1st verse line 3- youth that was
2nd verse line 1 (with) While
2nd verse line 3- delete away
2nd verse line 4- to my own eternal
2nd verse line 3- Start of quote; "I am going...
3rd verse line 2- Oh, down
3rd verse line 4- down just at noon (?)
5th verse line 3- But my sorrows will be over and I'll soon be forgot [Did he mean to put an extra line in the verse or did he just make a mistake and correct it?
7th verse line 3- end quote

(Sings better the way you transcribed it)