I'm glad Burton cut a lot of Sweeney Todd, as most of it doesn't work on film.
The Ballad: It's all exposition. And the smaller parts are used for scene changes. Cut!
A Little Priest: Most of it works because of the audience reaction.
Elixir & God, That's Good!: From reading the screenplay, it seems that Burton only wanted the main characters singing, so the ensemble parts are cut a bit. It's fine with me.
Green Finch: I think it's been trimmed down some. Which is fine, as I never got the song anyway.
I don't like what the screenwriter did with the ending. I think opening it with "No Place Like London" works. At least it works on paper. I don't mind the small line changes, either.
"Writing is like prostitution. First, you do it for love, then you do it for a few friends, and finally you do it for money." ~ Moliere