The entire song takes place before they've made any human pies. There is nothing in the text (or any production that I've ever seen) that would indicate that the entire song is not taking place in real time, all before they start making human pies. The song depicts the two of them joking about the kind of pies they can make.
Yes, in some productions, they actually do eat (or pretend eat) actual (or pretend) pies while they sing. But these are old pies Mrs. Lovett takes off her shelf.
Can anyone explain specifically what you are referring to that gives you the impression that time is moving forward in an accelerated way. And please don't say, "I've just always had that feeling." What in the text, or a production you've seen indicates that?
Art has a double face, of expression and illusion.