S.W.
Congrats on your nursing relationship!
I suspect the coughing is residual congestion that is loosened up by all the relaxing and swallowing involved with nursing. If the mucus in his nose is running down his throat, that could be irritating and cause the coughing, too. He would avoid sniffing and snuffling while nursing so he can breastfeed effectively, so perhaps it builds up and then he coughs to loosen it. The antibodies in your milk are helping to fight any infection that is trying to establish itself or hang around longer.
Best wishes!