The electron-electron interaction correction of first order in 1/Z to the one-electron part of the nuclear recoil effect on binding energies in atoms and ions is considered within the framework of the rigorous quantum electrodynamics (QED) approach. The calculations to all orders in αZ are performed for the 1s2 state in heliumlike ions and the 1s22s and 1s22p1/2 states in lithiumlike ions in the range Z=5-100. The results obtained are compared with the Breit-approximation values. The performed calculations complete a systematic treatment of the QED nuclear recoil effect up to the first order in 1/Z. The correction obtained is combined with the previously studied two-electron part as well as the higher-order electron-correlation corrections evaluated within the Breit approximation to get the total theoretical predictions for the mass shifts.