You don't clearly explain how the mother is not responsive, but I think it pretty odd that parents who just discovered their child has a developmental problem are insisting on leaving their child in the hands of someone who isn't licensed to care for special needs children. Even if there will be a specialist. You don't have the facilities or licensing for this.
Something just isn't adding up here. There is probably more than meets the eye. For sure, I agree with you that she is not being very up front with you about what's at issue with her son. I think its pretty odd that this mother is so insistent that you help her son, even though she supposedly sought out a professional evaluation. If the evaluation did indeed show that the boy is significantly developmentally behind as you describe, why didn't these same people suggest what the next course of action be? Wouldn't it be fair to guess that the people they went to for this evaluation would also suggest where this family should or could go to get further intervention and/or therapy? Considering the child is 6 months behind, wouldn't they have suggested a program that would address the needs of this child? Or is this delay something he'll grow out of and she's just trying to manipulate the situation to her benefit?
I do agree with others posts. If this child really does have special neeeds and you are not equipped or licensed to deal with that, then the obviously solution would be for you to tell them they need to seek appropriate daycare. Perhaps you could help them by tapping into some of your contacts and resources and pehaps put them in touch with a program or someone who specializes in special needs children.