We propose a realistic cold-atom quantum setting where topological localization induces nonreciprocal pumping. This is an intriguing non-Hermitian phenomenon that illustrates how topology, when assisted with atom loss, can act as a "switch" for the non-Hermitian skin effect (NHSE), rather than as a passive property that is modified by the NHSE. In particular, we present a lattice-shaking scenario to realize a two-dimensional cold-atom platform, where nonreciprocity is switched on only in the presence of both atom loss and topological localization due to time-reversal symmetry breaking. The resultant nonreciprocal pumping is manifested by asymmetric dynamical evolution, detectable by atomic populations along the system edges. Our setup may trigger possible applications in nonreciprocal atomtronics, where loss and topological mechanisms conspire to control atomic transport. Its quantum nature will also facilitate future studies on the interplay between non-Hermiticity and many-body physics.