What is the relation of the questioner to her half-brother's son from his mother's side—who is also her foster son—and is she his paternal aunt?

Suckling prohibits what lineage prohibits. So, whoever suckles from a woman becomes a son to her and to her husband, the owner of the milk. Therefore, your nephew through suckling is one of your mahrams, and you are his paternal aunt through suckling, even if the brother is only from the mother's side.