This paper investigates nonlinear fractional chemical-wave models governed by the Atangana–Baleanu–Caputo (ABC) fractional derivative. The motivation of this study arose from the need to model memory-dependent effects in nonlinear chemical-wave propagation, which cannot be adequately described using classical integer-order models. To solve the considered system, an $\alpha$-Laplace homotopy perturbation method ( $\alpha$-LHPM) was developed by combining the $\alpha$-Laplace transform with He's homotopy perturbation technique. The proposed approach provides recursive series solutions with rapid convergence and reduced computational complexity. Existence, uniqueness, and Hyers–Ulam stability results were also established under suitable assumptions. As an application, the fractional Belousov–Zhabotinsky dynamical system (BZDS) was analyzed for different fractional orders $0<\sigma \le 1$. Numerical simulations showed that decreasing the fractional-order parameter slows the wave propagation and produces smoother solution profiles due to stronger memory effects. In the classical case $\sigma =1$, the obtained numerical solutions showed excellent agreement with the exact solutions, with absolute errors of order ${10}^{-5}$ for $\zeta$ and ${10}^{-4}$ for $\omega$. These results demonstrate that the proposed $\alpha$-LHPM is an accurate and efficient semi-analytical tool for solving nonlinear fractional chemical-wave models.