"I don't think it is as calculated as you are all saying. I think it is an extension of the conversation the Judge initiates when he mentions that his "intended" is his ward. He starts to sing about generic Pretty Women when the judge questions Todd's statement "as pretty as her mother.""
Yes it's an extension of that conversation, but it's definitely still calculated. It's a very pointed number when you consider the nature of the judge's crimes, and Sweeney is fully aware of that.
"when Sweeney sang the 'Now Then My Friend' verse, he was going to kill Judge Turpin right away."
I think it's actually the opposite (unless I misunderstood you). If you listen to the lyrics of the "Now then my friend" verse, he's saying the exact opposite. He's going to go slowly, and enjoy himself. Lovett tells him to slow down and be patient, and "wait." When Sweeney is in the moment, he takes that advice. He tells himself "Patience, enjoy it. Revenge can't be taken in haste." After that, he blames Lovett, saying "Why did I wait? You told me to wait, now he'll never come again."