<p>This study investigates the unified boundary-layer flow of a non-Newtonian power law fluid induced by a stretched sheet or over a wedge. The rate of stretching is approximated in terms of a power law with respect to the edge of the boundary-layer. It is assumed that temperature has an effect on the fluid’s viscosity and thermal conductivity. The governing system of partial differential equations is transformed into a system of ordinary differential equations using suitable similarity transformation and analyzed using a Haar wavelets based computational scheme. The velocity ratio parameter unifies the two different physical flow scenarios, and a unified mathematical model is derived. The efficiency of the considered method is highlighted by describing the convergence rate and solution error for resolution levels. In addition, a comprehensive sensitivity analysis employing response surface methodology quantifies the influence of key parameters on critical engineering measures such as skin friction coefficient and Nusselt number. The results reveal intricate dependencies of thermal and momentum boundary layers on fluid rheology and property variations, offering valuable insights for the design and optimization of industrial processes involving complex fluid flows. Major findings include: (a) the power-law index significantly influences both thermal and velocity boundary-layer thickness (b) increased temperature dependence of viscosity or thermal conductivity leads to notable changes in flow and heat transfer rates and (c) the velocity ratio parameter dominates the sensitivity of engineering responses, a result of direct practical significance.</p>