I have had this situation come up before and I just handled it by talking to the leaving family or in one case "suspected leaving" family. Both times it worked out well and they accommodated me. They also want to keep in good graces with me if they want me to take their next child.

I personally would tell them that I have a family for October that wants the spot and see what they would say. I wouldn't have a hard time telling them that their end date will be end of September if it was when they said they were going to go anyways. If they needed until Oct 15th I would see if the other family could accommodate this (I find most do if they really want the spot). If they can't then I let the first family know that unfortunately this other family is taking the spot Oct 1st and you are telling them now so they can make alternative plans.