Neutron star mergers (NSMs) eject energetic subrelativistic dynamical ejecta into circumbinary media. Analogous to supernovae and supernova remnants, the NSM dynamical ejecta are expected to produce nonthermal emission by electrons accelerated at a shock wave. In this paper, we present the expected radio and X-ray signals by this mechanism, taking into account nonlinear diffusive shock acceleration (DSA) and magnetic field amplification. We suggest that the NSM is unique as a DSA site, where the seed relativistic electrons are abundantly provided by the decays of r-process elements. The signal is predicted to peak at a few 100-1000 days after the merger, determined by the balance between the decrease of the number of seed electrons and the increase of the dissipated kinetic energy, due to the shock expansion. While the resulting flux can ideally reach the maximum flux expected from near-equipartition, the available kinetic energy dissipation rate of the NSM ejecta limits the detectability of such a signal. It is likely that the radio and X-ray emission are overwhelmed by other mechanisms (e.g., an off-axis jet) for an observer placed in a jet direction (i.e., for GW170817). However, for an off-axis observer, to be discovered once a number of NSMs are identified, the dynamical ejecta component is predicted to dominate the nonthermal emission. While the detection of this signal is challenging even with near-future facilities, this potentially provides a robust probe of the creation of r-process elements in NSMs.