Nanozymes (NZs) are nanostructured artificial enzymes that mimic catalytic properties of natural enzymes. The NZs have essential advantages over natural enzymes, namely low preparation costs, stability, high surface area, self-assembling capability, size and composition-dependent activities, broad possibility for modification, and biocompatibility. NZs have wide potential practical applications as catalysts in biosensorics, fuel-cell technology, environmental biotechnology, and medicine. Most known NZs are mimetics of oxidoreductases or hydrolases. The present work aimed to obtain effective artificial peroxidase (PO)-like NZs (nanoPOs), to characterize them, and to estimate the prospects of their analytical application. NanoPOs were synthesized using a number of nanoparticles (NPs) of transition and noble metals and were screened for their catalytic activity in solution and on electrodes. The most effective nanoPOs were chosen as NZs and characterized by their catalytic activity. Kinetic parameters, size, and structure of the best nanoPOs (Cu/CeS) were determined. Cu/CeS-based sensor for H2O2 determination showed high sensitivity (1890 A·M−1·m−2) and broad linear range (1.5–20,000 µM). The possibility to apply Cu/CeS-NZ as a selective layer in an amperometric sensor for hydrogen-peroxide analysis of commercial disinfectant samples was demonstrated.