This study numerically investigates dual-phase thermal and concentration relaxation effects on magneto-viscoelastic thermally radiating Williamson nanofluid flowing over a permeable stretching surface in 2D and steady-state conditions using the Cattaneo–Christov flux model with consideration of the effects of viscous dissipation and joule heating. The governing PDEs have been first transformed into a highly nonlinear coupled ODEs via suitable similarity variables. The resultant equations are solved with a two-step high-accuracy spectral quasilinearization method, whose accuracy and convergence are rigorously validated against previously published results. This allows for an in-depth investigation of how the key parameters including the Weissenberg number, velocity ratio, suction/injection, magnetic field, radiation, and relaxation times influence the characteristics of the boundary layer. Results show that thermal relaxation suppresses heat transfer by delaying thermal diffusion, while concentration relaxation enhances mass transfer through sharper concentration gradients. The velocity ratio increases skin friction drag, velocity, and rate of heat flow but reduces rate of mass flow; suction thins boundary layers to promote heat transfer, whereas injection amplifies velocity and temperature profiles, further boosting heat transfer. Viscoelasticity lowers skin friction and mass transfer but can enhance heat transfer, and magnetic and viscous dissipation effects further improve mass transfer. This work demonstrates the superiority of the Cattaneo–Christov model over classical Fourier and Fick laws in capturing finite-speed relaxation phenomena, offering critical insights for optimizing thermal and mass transport in engineering applications involving non-Newtonian nanofluids.