The current-convective instability is one of the physical mechanisms that can drive fluctuating transport of plasma in the tokamak divertor. The impact of the tokamak magnetic geometry on spatial and temporal properties of saturated turbulence, driven by the instability, is investigated. The analysis is performed by employing the basic model of the instability [A. A. Stepanenko and S. I. Krasheninnikov, Phys. Plasmas 25, 012305 (2018)] extended to include contributions from the interchange drive. Turbulence simulations are carried out in BOUT++ under DIII-D-like conditions. The dependencies of amplitudes, spatial and frequency spectra of saturated plasma fluctuations on the curvature radius, and connection and shear lengths of magnetic field lines are found. It is demonstrated that taking the finite curvature of the magnetic field into account leads to a large increase in the turbulence amplitude, whereas the shear and connection lengths mainly affect the temporal dynamics of fluctuations and their spatial localization. In all cases shown, the turbulence frequency and wave-number spectra demonstrate better agreement with the experimentally observed ones, compared to the results of previous simulations based on the basic model of the current-convective instability.