I would believe the child when he said that he didn't go to the doctors but that doesn't mean that the mom never took him at all. Maybe she took him 2 weeks ago or prior and the doctor mentioned it was allergies at that time.
We have had a terrible cough go through the daycare this year that started in September and it seems one kid or another has it at any given time.
I went to the doctor for it three times, had xrays,antibiotics and finally puffers . The doctor said it was likely allergies and that I would just need puffers until it warms up. So...her claim of the cough being related to allergies could be entirely true. I have never had anything like this or had an allergy-related cough and I wouldn't have believed such a thing was possible until I heard it this year.
I have personally never chosen to exclude a child from daycare due to a cold/cough.