The electronic properties of heterojunction electron gases formed in GaN/AlGaN core/shell nanowires with hexagonal and triangular cross sections are studied theoretically. We show that at nanoscale dimensions, the nonpolar hexagonal system exhibits degenerate quasi-one-dimensional electron gases at the hexagon corners, which transition to a core-centered electron gas at lower doping. In contrast, polar triangular core/shell nanowires show either a nondegenerate electron gas on the polar face or a single quasi-one-dimensional electron gas at the corner opposite the polar face, depending on the termination of the polar face. More generally, our results indicate that electron gases in closed nanoscale systems are qualitatively different from their bulk counterparts.