If a person with allergies,sneezes into a public pool, will this cause illness? My husband had pneumonia and I worry about this reoccurring?

No, that will not cause illness unless the person sneezing is suffering a bacterial or viral illness. At that point the chlorine should kill the virus or bacteria.

Agree, allergy is not an infection, and not something you "catch" more concern would be over touching door handles, and crowded elevators where people don't cover their mouth and so on.