Carbon nanotube (CNT)/copper composites offer promise as lightweight temperature-stable electrical conductors for future electrical and electronic devices substituting copper. However, clarifying how constituent nanotube structures influence CNT/Cu electrical performances has remained a major research challenge. Here, we investigate the correlation between the CNT/Cu electrical performances and nanotube structure by preparing and characterizing composites containing nanotubes of different structural attributes. We prepared three types of composites—single-wall (SW)-CNT/Cu wires, SW-CNT/Cu pillars, and multi-wall (MW)-CNT/Cu wires. The composites were fabricated from the corresponding CNT templates by two-step Cu electrodeposition, which retains template nanotube attributes through the fabrication process. The nanotube characteristics (diameter, G/D, alignment, etc.) in each template as well as the internal structure and electrical performances of the corresponding composites were characterized. SW-CNT/Cu wires and pillars outperformed MW-CNT/Cu wires, showing ≈ 3× higher room-temperature four-probe conductivities (as high as 30–40% Cu-conductivity). SW-CNT/Cu also showed up to 4× lower temperature coefficients of resistances i.e., more temperature-stable conductivities than MW-CNT/Cu. Our results suggest that few-walled small-diameter nanotubes can contribute to superior temperature-stable CNT/Cu conductivities. Better CNT crystallinity (high G/D), fewer nanotube ends/junctions, and nanotube alignment may be additionally beneficial. We believe that these results contribute to strategies for improving CNT/Cu performances to enable the real-world application of these materials as Cu substitutes.