Agee &co kept it in at least up to the shooting draft of the screenplay:

"Against this, o.s., mostly in breathy, lonesome undertone, on one or two phrases loudly and assertively, Allnutt is singing, rottenly and inchoately, some part of the following:

[ALLNUTT'S VOICE, (o.s.; singing)]
Gimmy regards ter Leicester Square
Sweet Piccadilly an' Myefair,
Remember me to the folks darn there
They'll under-sta-and.

SLOW FADE on Rose as first daylight begins to appear.

[Agee, James; Huston, John; Viertel, Peter; The African Queen, screenplay, 1951]