A magnetic toroidal moment is a fundamental electronic degree of freedom in the absence of both spatial inversion and time-reversal symmetries and gives rise to novel multiferroic and transport properties. We elucidate essential model parameters of the nonlinear transport in the space-time (PT) symmetric collinear antiferromagnetic metals accompanying a magnetic toroidal moment. By analyzing the longitudinal and transverse components of the second-order nonlinear conductivity on a two-dimensionally stacked zigzag chain based on the nonlinear Kubo formula, we show that an effective coupling between the magnetic toroidal moment and the antisymmetric spin-orbit interaction is an essential source of the nonlinear conductivity. Moreover, we find that the nonreciprocal longitudinal current and nonlinear transverse current in a multiband system are largely enhanced just below the transition temperature of the antiferromagnetic ordering. We also discuss the relevance of the nonlinear conductivity to the linear magnetoelectric coefficient and conductivity. Our result serves as a guide for exploring the microscopic essence and clarifying the parameter dependence of the nonlinear conductive phenomena in ferrotoroidal metals.