Nanoscale nonreciprocal elements are needed to control signal flow in scalable quantum technologies, but existing nonreciprocal components are often bulky. Here, we show that a strongly nonreciprocal current-voltage relation can arise entirely from coherent elastic scattering in a mesoscopic Rashba nanowire with gate-defined scatterers. We first identify the symmetry constraints that enforce reciprocal transmission in a phase coherent conductor even when inversion and time-reversal symmetry are broken. We then show how gates that are capacitively coupled to the leads, together with a magnetic scatterer or transverse subband mixing, can remove these constraints. In the weak-scattering regime, nonreciprocal conductance originates from interference between distinct backscattering paths. In the strong-scattering regime, nonreciprocity can be resonantly enhanced by the hybridization of the transmitting channel with bound states that arise from a higher transverse subband or spin-split band. In this regime, we find that a bias-induced change in the gate potentials of about one percent can produce an order-unity diode response, i.e., electrons are transmitted almost as in a clean wire for one bias polarity, whereas for the other polarity the current is strongly suppressed.