Child's playmate has persistent cough...?

Please understand that you and your child may or may not be at risk. You need to call a doctor or clinic as soon as possible. Tell them what you have said here about both the neighbor child's cough and the mother's history of TB (tuberculosis). Then set up an appointment to get tested for TB. TB is very serious and can be fatal if not treated. Everything you have said here points to an infection of some sort, not necessarily to TB, but not to allergies either. Allergies do not cause persistent cough or fever, and are not treated with antibiotics. If it turns out that you have been exposed to TB, follow the doctor's orders for treatment and you should be fine. Also, the county or state health department will contact your neighbors and work to take care of that child.