I love that musical so much.

Sweeney- INFP. His Fi is very strong, and ISFP doesn't seem like a good fit to me.
Mrs. Lovett- ENTJ. Clearly an extrovert, always scheming, uses Fi and Se, has no empathy for the people around her other than Toby and Sweeney.
Judge Turpin- ESTJ, pretty clear.
Adolfo Pirelli- ESTP, or ENTP? Maybe it's just Sacha Baron Cohen that I have in mind after the movie.
Johanna- haven't seen much of her. ISFP, I think; wants to be free.
Anotony- uses Fe. EXFJ?
Toby- ISFJ. Seems introvert, very true. Very loyal.
Beadle- ISTJ, maybe.
Lucy- no clue.