Singers often vary the tune to suit their voice and their taste. The Fifes noted a similarity in part to "The Waggoneer's Lad," which I have heard used. I have heard some with a good upper range move parts of some couplets up to give variety while others are kept almost to a low monotone.
I just looked at three, Lomax, ABFS, Fife & Fife and Jules Allen. The waltz tempo is the same, but emphasis, and therefore 'feel,' are different. Just looking at three words- In Verne, in the phrase 'my pony won't stand,' po of pony is very low while 'ny' is high and glides into a descending two-note 'won't. In Lomax, 'po' is high and 'ny' drops slightly, to rise again on 'won't.' In Fife, all are middle to low scale.

Not having seen the film, I don't know what tune or tune variance was used. The music arranger may have used a new tune altogether.