I would do exactly this. Begin looking for someone to fill the spot. Give mom an exact date that you will require an answer by, and if you haven't heard, I would have to say that means she is leaving.
You don't have to be rude or angry, but she can't just leave you twisting in the wind here. Otherwise you will find that she gives you notice, and you're left scrambling to fill the space. I suppose in that way, it's good that it's only a p/t slot. Based on what I am seeing, I don't see how they could stay, if she is that allergic. Only one course of action... sad though it may be for you to say goodbye.
Good luck, my dear!!!