Rather than commenting on each of your posts about Chaplin's voice, I thought I'd throw out my take on its significance. I'm interested in the fact that most of you found this scene a little disappointing, even impenetrable.
For me, the important point here is that the song, even though it sounds foreign, doesn't really make sense in any language. Your sense that it is nonsense is correct - hence the confusion and mild disappointment.
But here's the question: why do we have all this buildup towards nonsense, and why does Chaplin get to utter nonsense in a film which more or less forbids people to speak? Answering that question is pretty central to the film.
My point of view is that the film is arguing that in the age of mechanical reproduction* (of voice, of culture, of everything) only voices of authority (e.g., the boss's voice and the radio) are actually heard -- other voices, being merely organic, appear as nonsense.
Or, to put a more positive spin on it - nonsense is the only alternative to submission, the only way of rebelling against the system. Chaplin isn't much of a rebel - but on the other hand he's all we've got in the context of the film.
* I'm drawing here on Walter Benjamin's essay "The Work of Art in the Age of Mechanical Reproduction," which was published in the same year as Modern Times, 1936.