An interface between LaAlO3 and SrTiO3 (LAO/STO) may host two-dimensional superconductivity, strong spin-orbit coupling, and magnetic ordering. There are indications that the interfacial superconducting layer is nonhomogeneous at the nanoscale. The presence of inhomogeneities may have strong implications on the superconducting transport properties and lead to formation of weak links between superconducting regions. In this paper, we report on the temperature and magnetic field dependence of superconducting transport in single nanowires and nanorings fabricated in a LAO/STO interface. The analysis of data proves that our nanostructures behave like uniform superconducting filaments carrying a critical current close to the theoretical Ginzburg-Landau depairing limit. Furthermore, we unexpectedly find an enhancement of the critical current of LAO/STO nanowires and nanorings at low temperature when a magnetic field is applied. Our results point towards the coexistence of a homogeneous interfacial superconductivity and spatially separated magnetism at the LAO/STO interface.