In evolved supernova remnants (SNRs) interacting with molecular clouds, a highly inhomogeneous structure consisting of a forward shock of moderate Mach number, a cooling layer, a dense radiative shell, and an interior region filled with hot tenuous plasma is expected. We present a model of nonthermal electron injection, acceleration, and propagation in that environment and find that these SNRs are efficient electron accelerators and sources of hard X-ray and -ray emission. A forward shock of velocity vs 100 km s-1 with an ionized precursor propagating into the molecular cloud accompanied by magnetohydrodynamic turbulence provides a spatially inhomogeneous distribution of nonthermal electrons. The energy spectrum of the nonthermal electrons is shaped by the joint action of first- and second-order Fermi acceleration in a turbulent plasma with substantial Coulomb losses. Bremsstrahlung, synchrotron, and inverse Compton radiation of the nonthermal electrons produce multiwavelength photon spectra in quantitative agreement with the radio and hard emission from IC 443 observed by ASCA and EGRET. We distinguish interclump shock wave emission from molecular clump shock wave emission; particles reach higher energies in the interclump shock, and that is the likely source of -ray emission and radio synchrotron emission. Spatially resolved X-ray and -ray spectra from the SNRs IC 443, W44, and 3C 391 as might be observed with BeppoSAX, Chandra XRO, XMM, International Gamma-Ray Astrophysical Laboratory, and Gamma-Ray Large Area Space Telescope would distinguish the contribution of the energetic lepton component to the -rays observed by EGRET, constraining the cosmic-ray nuclear component spectra in these SNRs. These data would provide a valuable tool for studying the complex structure of molecular clouds where SNR radiative shocks interact with dense molecular clumps.