Presently, the main cause of pollution of natural water resources is heavy metal ions. The removal of metal ions such as nickel (Ni2+) and cadmium (Cd2+) has been given considerable attention due to their health and environmental risks. In this regard, for wastewater treatment containing heavy metal ions, graphene oxide (GO) nanocomposites with metal oxide nanoparticles (NPs) attained significant importance. In this study, graphene oxide stacked with copper oxide nanocomposites (GO/CuO-NCs) were synthesized and characterized by Fourier transform infrared (FTIR), X-ray diffraction (XRD), scanning electron microscopy (SEM), energy dispersive X-ray (EDX), and atomic force microscopy (AFM) analytical procedures. The prepared GO/CuO-NCs were applied for the removal of Ni2+ and Cd2+ ions from a binary metal ion system in batch and continuous experiments. The obtained results revealed that GO/CuO-NCs exhibited the highest removal efficiencies of Ni2+ (89.60% ± 2.12%) and Cd2+ (97.10% ± 1.91%) at the optimum values of pH: 8, dose: 0.25 g, contact time: 60 min, and at 50 ppm initial metal ion concentration in a batch study. However, 4 mL/min flow rate, 50 ppm initial concentration, and 2 cm bed height were proved to be the suitable conditions for metal ion adsorption in the column study. The kinetic adsorption data exhibited the best fitting with the pseudo-second-order model. The adsorption isotherm provided the best-fitting data in the Langmuir isotherm model. This study suggested that the GO/CuO nanocomposites have proved to be efficient adsorbents for Ni2+ and Cd2+ ions from a binary metal system.