  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 QED approach. The
calculations to all orders in $\alpha Z$ are performed for the $1s^2$ state in
heliumlike ions and the $1s^2 2s$ and $1s^2 2p_{1/2}$ 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.
