Spoiler ahead.
(I know Movie adaptations of plays may have different interpretations as that of the original, but this thought just can't rest in my head)
Though I am a fan of Tim Burton, I don't think he established the presence of Lucy as the beggar woman in the movie quite well.
In the original broadway musical, Lucy (as the beggar woman) already appeared in the early part of the play, right before Todd sings "The Barber and his Wife". Her presence before Todd introducing her in the play already showed Lucy's presence without Todd's knowing. In the movie, this scene was cut; Todd sang "The Barber and his Wife" and Lucy (as the beggar woman) was not present. She only appeared in "Ah, Miss", where Anthony has come to know about Johanna and Judge Turpin.
The scene I have mentioned was, in my opinion, needed. Though it wasn't explicitly shown in the play, Lucy was already present to the audience, given her line "Hey, don't I know you, Mister?" that was recurring in the play. The presence of the scene could have justified the effect of Todd's revenge in the story. The line was only said once in the movie, only after Beadle Bamford was killed.
The loss of the scene would have been compensated for the film's advantage of showing the audience images that may have given the same effect. Similar to the play, the actress playing the beggar woman is the same as that of Lucy. There were parts in the film that did close up to her face, but this was seldom. Her character as the crazy beggar woman was shown more than that of her being Lucy. The only time her face as closed up what shen she said "Hey, don't I know you, Mister?" (See? it does give the audience the idea of her possibility being Lucy.)
Though the ending gave the same aura as that of the play, if you're going to find parts in the film that would justify the ending, there are hardly any that would justify it. (I don't consider Mrs. Lovett's explanation sufficient. The line "Said she took a poison, she did. Never said that she died." just made Todd, as well as the audience, seem like fools more than beng a clue of Lucy's presence.)