We investigate spontaneous and pumped entanglement of two-level systems in the vicinity of a photonic topological insulator interface, which supports a nonreciprocal (unidirectional), scattering-immune, and topologically protected surface-plasmon polariton in the band gap of the bulk material. To this end, we derive a master equation for qubit interactions in a general three-dimensional, nonreciprocal, inhomogeneous, and lossy environment. The environment is represented exactly, via the photonic Green's function. The resulting entanglement is shown to be extremely robust to defects occurring in the material system, such that strong entanglement is maintained even if the interface exhibits electrically large and geometrically sharp discontinuities. Alternatively, depending on the initial excitation state, using a nonreciprocal environment allows two qubits to remain unentangled even for very close spacing. The topological nature of the material is manifest in the insensitivity of the entanglement to variations in the material parameters that preserve the gap Chern number. Our formulation and results should be useful for both fundamental investigations of quantum dynamics in nonreciprocal environments and technological applications related to entanglement in two-level systems.