A heat pipe cooled reactor is a new type of nuclear reactor that utilizes phase-change heat transfer between the evaporation and condensation sections of the working fluid within heat pipes to achieve passive conduction of core heat. This eliminates the need for pumps and valves, simplifying system design. The reactor leverages the efficient heat transfer characteristics of heat pipes to quickly transfer high-temperature heat generated in the core to the hot end of a thermoelectric generation module, while the cold end dissipates heat through radiative heat pipes. Thermoelectric generation technology directly and efficiently converts thermal energy into electrical energy using the temperature difference between the core and the cooling system. In this paper, we have developed a numerical simulation code targeting the coupled heat transfer process of the “reactor core–high-temperature heat pipe–thermoelectric generation simulator–heat pipe radiative cooling system,” aiming to study its startup characteristics. The code integrates a core high-temperature sodium heat pipe module, a thermoelectric generation simulator, and a finned radiative potassium heat pipe module, achieving coupled heat transfer calculations among these modules. To verify the accuracy of the code, we compared the calculated results with experimental data from a non-nuclear test prototype of a heat pipe cooled space reactor. The code effectively calculates the temperature variation trend during system startup with minimal errors, demonstrating its predictive accuracy. This work can provide a reference for the design and optimization of heat pipe cooled space reactors.