CD8⁺T cells are virus-clearing core effector cells, and their response is regulated by viral latency and cell spatial migration, the former causing time delay and the latter causing spatial non-uniformity. Most existing models have dealt with these two factors in isolation, making it difficult to explain the coexistence of viral load oscillations and foci. This paper constructs a CD8⁺T cell antiviral response diffusion model coupled with time delay and diffusion, and analyzes its nonlinear dynamic behavior. Theoretical analysis shows that delayed viral replication and immune activation trigger Hopf bifurcation to oscillate viral load; The mismatch between the effect and the diffusion coefficient of the infected cells triggered the Turing bifurcation, forming a spatial pattern. In the overlapping area of parameters, delay and diffusion coupling produce behaviors such as traveling waves, essentially a double breaking of spatiotemporal translational symmetry. Numerical simulations verify the bifurcation critical conditions and give predictions. This study provides a unified framework for understanding the immune response of CD8⁺T cells and has reference value for infection prognosis and treatment.