Buoyancy-driven viscous fluid flow across a curved surface is investigated numerically in this work using the coupled Maxwell and Navier–Stokes equations, with variable fluid characteristics represented as nonlinear functions of temperature. Realistic magneto-hydrodynamic effects are captured by including the Lorentz force and the influence of a fluctuating magnetic field in curvilinear coordinates. The governing partial differential equations are solved using the parametric continuation method (PCM) after being converted into a system of ordinary differential equations by similarity transformations. Results demonstrate excellent agreement when compared to previously published data using MATLAB's PCM solver to confirm correctness. According to the parametric study, buoyancy ((Formula presented.)) improves fluid motion by around 15%, whereas greater curvature factors (Formula presented.), Stuart numbers (Formula presented.), and Prandtl numbers (Formula presented.) result in a 12%–16% drop in radial and arc-length velocities. The temperature profile falls by more than 23% as (Formula presented.) and (Formula presented.) increase, indicating the significance of thermal diffusivity in preventing heat buildup. It increases by 25% with higher magnetic interaction ((Formula presented.), (Formula presented.)). The induced magnetic field is strengthened by 6%–7% with a little increase in the magnetic interaction parameter (Formula presented.), whereas the magnetic field intensity is reduced by about 25% with a larger (Formula presented.). Skin friction falls by almost 10% with greater (Formula presented.) at moderate (Formula presented.), but increases by 4% under larger Lorentz forces ((Formula presented.), (Formula presented.)). Overall, the results show that velocity, temperature, magnetic field distribution and surface forces are strongly influenced by buoyancy, curvature and electromagnetic parameters. The findings shed light on efficient energy optimisation, thermal control, and electromagnetic regulation of MHD flows over curved geometries.