"Little Lord Fauntleroy"(I think my spelling is way off on this one). It was a made for TV miniseries that starred a very young Ricky Schroeder (pre-SilverSpoons).
I remember seeing it the first time when I was little; I'd like to say it was sometime around christmas. Anyway, the story was very touching. I probably watched it about a hundred times when I was little.
I believe that song was used to symbolize that even a peasant American boy could have something to offer in a royal English court. That was the basic gist of the movie. That is probably why the somg was used so much.