Structure and evolution kinetics of non-equilibrium clusters formed in a solution of oppositely charged nanoparticles are studied using a recently developed kinetic Monte Carlo simulation scheme (Jha et al., Soft Matter, 2012, 8, 227-234). A diverse range of dynamic cluster configurations are obtained by varying the interaction strength between nanoparticles, screening length, and packing density of nanoparticles. Structural details of the resulting clusters are obtained using the correlations of local bond orientational order parameters. At low-salt concentrations (weak screening), clusters with structures ranging from NaCl-type cubic aggregates to fibril-like chains are observed, while at high-salt concentrations (strong screening), disordered compact clusters are observed. A chain-folding barrier model is proposed to explain the kinetically trapped fibril-like assemblies. In higher-density solutions, large ionic clusters or percolated gel structures are observed. Our work demonstrates the structural richness of non-equilibrium ionic assemblies of oppositely charged nanoparticles and elucidates the effect of ion correlations on the determination of the structure of assemblies of oppositely charged nanoparticles. These "nanoionic composites" hold great promise in a variety of emerging applications such as templated polymerization of charged molecules and assembly of charged nano-objects.