Thermally conductive phase-change materials (PCMs) were produced using the crosslinked Poly (Styrene-block-Ethylene Glycol Di Methyl Methacrylate) (PS-PEG DM) copolymer by employing boron nitride (BN)/lead oxide (PbO) nanoparticles. Differential Scanning Calorimetry (DSC) and Thermogravimetric Analysis (TGA) methods were used to research the phase transition temperatures, the phase-change enthalpies (melting enthalpy (ΔHm), and crystallization enthalpies (ΔHc)). The thermal conductivities (λ) of the PS-PEG/BN/PbO PCM nanocomposites were investigated. The λ value of PS-PEG/BN/PbO PCM nanocomposite containing BN 13 wt%, PbO 60.90 wt%, and PS-PEG 26.10 wt% was determined to be 18.874 W/(mK). The crystallization fraction (Fc) values of PS-PEG (1000), PS-PEG (1500), and PS-PEG (10,000) copolymers were 0.032, 0.034, and 0.063, respectively. XRD results of the PCM nanocomposites showed that the sharp diffraction peaks at 17.00 and 25.28 °C of the PS-PEG copolymer belonged to the PEG part. Since the PS-PEG/PbO and the PS-PEG/PbO/BN nanocomposites show remarkable thermal conductivity performance, they can be used as conductive polymer nanocomposites for effective heat dissipation in heat exchangers, power electronics, electric motors, generators, communication, and lighting equipment. At the same time, according to our results, PCM nanocomposites can be considered as heat storage materials in energy storage systems.