I do not agree with negative characterization the willingness of public school teachers to write recommendation letters. I have one kid in private school and another one in public school who has applied to private schools this year. The teachers have been most gracious both times, although one needs to be respectful of their other responsibilities too. Like with anything I think respect and politeness goes a long way. With so many kids applying one of the public school teachers told us she had to write 40 letters for her students, so we made sure to make sure she had plenty of advance notice, and some polite reminders about the deadline along the way. I think if kids can understand that the world doesn’t “owe” them anything…but if they want something…they have to work for it…they could really benefit. I do not know what exactly has happened between the poster and the teacher…but my advice will be to think about it and mend anything you can at your end to improve your relationship.
@Sush2016: Guess it depends on the teacher. My kid needed several for applications. Asked early and all were received. But I cannot say that they were all nice 3/4 isn’t bad. I don’t think kids or parents should have to grovel for what is clearly something very necessary to move into a private school. ( Yes, we asked early, made it as easy as possible etc etc). In our town they are mandated by the teachers union contract. (Thank God) Teachers do very little outside of the contract. You might think that is ok and maybe it is. But if you are a kid and ask well in advance and nicely and give a self addressed stamped envelope and follow all of the rules, you should also not be subject to the whims of a teacher/admin who thinks “they are doing you a favor” and make you grovel. I have seen it. Don’t like it but in our public school that’s the norm. I think kids are due a written review. Sadly, I wish the schools allowed kids to chose the subject. Then one could avoid these teachers. But the kids are stuck. To those who live in public school districts where the teachers willingly write letters, that’s great. Kids should always be respectful and teachers should always treat kids with respect as well. You have to have experienced it to be on the receiving end. Else, how do you know how some kid feels who cannot get a math recommendation because a teacher “fill in the blank” doesn’t feel like it, thinks kid should ----, will write only x number, etc. Silly really. Why should the teacher have ANY say in something so important to a child’s future. Answer they shouldn’t.
I think the main point is that the teacher agreed to do it, then changed their mind (apparently some weeks later, after the deadline passed). That should be taken to the GC and/or principal, IMHO. But you are going to have to get another teacher. Even if they “make” the teacher do it, it now won’t be a good letter.
I’d approach the new teacher very apologetically. Tell them you know it is very late, but that another teacher had agreed to write previously but now says they don’t have time. I would NOT name the other teacher. – just say you don’t feel comfortable doing so if they press you. They can probably find out, but I wouldn’t go into details at all on interactions with the other teacher. Just ask if they can please help you, and ask if they can please do it within a week because the date is now passed. Offer to give them any info they need to help them with the letter.
What else can you really do at this point?
I don’t know of any college that insists on “current teachers”. That is silly. Letters are often written in October or Nov when the teacher has only had the kid for a month or two in class. Generally it is best to get letters from junior year teachers who know you well and have good things to say about you. A senior year teacher that you feel does know you and has great things to say is another option. In a true crunch… a sophomore year teacher will do, but I wouldn’t do it for top schools. Maybe if they were an AP class teacher and you were a terrific student in that class.
I agree with you on colleges, @intparent, but this is for prep schools and they ask for current year teachers.
This thread makes me appreciate even more the fine and dedicated public middle school teachers my kids had and the counseling office that helped orchestrate the filing of paperwork. They were all very supportive of my kids when they applied to private schools.
Ah. Sorry! Caught by the “Lastest Posts” trap. 
I still think the OP just needs to ask another teacher, and explain that another teacher had agreed to, but now does not have time.