<p>The study of nanofluid transport dynamics has garnered significant attention in recent years due to its vast potential in enhancing heat and mass transfer in engineering and biomedical applications. In particular, the interaction of gyrotactic microorganisms and thermophoretic forces within two-phase nanofluid systems introduces novel mechanisms for controlling flow stability and thermal efficiency. This research investigates a two-phase model of Carreau and Casson fluids transporting nanoscale particles to analyze the flow and thermal performance over static wedge (SW) and moving wedge (MW) surfaces. The study examines transport phenomena involving gyrotactic microorganisms and the movement of tiny particles influenced by thermophoresis and Brownian motion, especially when fluid concentration significantly affects particle migration. The dimensionless ordinary differential equations are solved using the Wavelets and Gegenbauer wavelets techniques with the help of Mathematica 11.3 software and provide a deep discussion. The findings demonstrate that Casson liquids exhibit greater wall shear stress coefficients in the SW surface, while Carreau liquids achieve the highest heat transfer rate, mass transfer rate, and local motile microorganism concentration in MW surface. These findings contribute valuable knowledge to the design and optimization of advanced thermal systems, paving the way for improved industrial efficiency and sustainable energy solutions.</p>