This study investigates the combined effects of Soret and Dufour phenomena on the hydrothermal transport of an alumina–water couple stress nanofluid, modeled using the Buongiorno nanofluid framework that incorporates Brownian motion and thermophoretic diffusion mechanisms. The fluid motion is considered over a curved oscillatory stretching surface, representing an important class of configurations in industrial and biomedical heat transport applications, including cooling technologies, thermal energy storage, and advanced process engineering systems. The governing model is established through a system of coupled nonlinear partial differential equations (PDEs) describing momentum, energy, and nanoparticle concentration fields. Nonlinear radiative heat transfer is incorporated via the Rosseland approximation to capture the influence of high-temperature thermal radiation. Suitable similarity transformations are used to transform the governing flow equations into a nonlinear dimensionless set of partial differential equations. The obtained equations are then solved analytically by incorporating the homotopy analysis method (HAM) to establish high accuracy and convergence control. Parametric examination reveals that the amplitude of velocity decreases with increasing the magnetic parameter, radius of curvature, nanoparticle volume fraction, and couple stress parameter. Furthermore, the temperature distribution increases with higher Dufour number and nanoparticle volume fraction. In general, the current study highlights the noteworthy interaction of coupled transport phenomena and alumina nanoparticles on flow, heat, and mass transfer characteristics. The novelty of this study lies in the simultaneous consideration of nonlinear radiation, Soret–Dufour effects, and curved oscillatory stretching surfaces in a couple-stress nanofluid, providing a new insight beyond existing literature for advanced thermal management system design.