The influence of local and nonlocal nucleon-nucleon (NN) potential models on the unpolarized structure functions $A(Q^2)$ and $B(Q^2)$ and the deuteron electromagnetic form factors in the elastic electron-deuteron (e-d) scattering process is investigated. As local NN potentials, the realistic Argonne v18 and chiral NNNLO potentials are used. For the nonlocal ones, the modern CD-Bonn and Nijmegen-I potentials are considered. Various parameterizations of the free nucleon form factors are used to study the sensitivity of the predicted results to the nucleon structure. We found that the structure functions $A(Q^2)$ and $B(Q^2)$ as well as the charge $G_C(Q^2)$, quadrupole $G_Q(Q^2)$, and magnetic $G_M(Q^2)$ deuteron form factors are slightly dependent on the NN potential model used for the deuteron wave function. On the contrary, the results of structure functions and form factors are found to be strongly dependent on the models used for the free nucleon form factors.