The quantum Cheshire cat effect is an important phenomenon in quantum mechanics that reveals the separability of physical properties from their carriers. This effect transcends the classical framework whose attributes must be inherently attached to objects, providing new perspectives for quantum information and precision measurement. Based on the quantum Cheshire cat effect, we prepare a pre-selected state of a spin-${1}/{2}$ atomic system composed of two particles through a pre-selection process. We conduct quantum weak measurements on the spins and positions of these two atoms and extract weak values using the method of imaginary time evolution (ITE)(Fig. (1)). Subsequently, we perform post-selection on these two atoms, designing two distinct post-selected states. Initially, we calculate analytical solutions when both atoms encounter these two different post-selected states separately. During the stage of obtaining weak values via ITE, we discuss the scenario with only one post-selected state. In this case, our experimental goal is to achieve spin exchange between the two atoms. We apply ITE to obtain normalized coincidence rate for the system. By performing linear fitting on these normalized coincidence rates, we derive numerical solutions for the system's weak values. A comparison between analytical and numerical solutions indicates that they are in close agreement, demonstrating that our method makes spin exchange between the two atoms. Next, we examine scenarios involving both post-selected states during post-selection. After completing weak measurements on particles, delayed-choice allows them to evolve along different paths ultimately leading to distinct post-selected states. One particular post-selected state resulted in final measurement outcomes indicates that spin exchange occurs between both particles with amplification. Conversely, the other post-selected state ensures that, even after undergoing weak measurement and delayed-choice, the states of the two particles remain consistent with their pre-measurement conditions. We also compare the analytical and numerical solutions of the experiment involving delayed-choice and find them to be largely consistent. This consistency indicates that delayed-choice indeed has a significant impact on whether the final exchange occurs. Our research theoretically confirms the feasibility of fermionic systems within bipartite quantum Cheshire cat effects and illustrates how delayed-choice influences quantum Cheshire cat effects in spin-${1}/{2}$ atomic systems.