This study rigorously investigates the flow dynamics of hybrid nanofluids comprising Cu + Al2O3 nanoparticles in Ethylene Glycol around a cylindrical shape. The analysis incorporates advanced concepts such as non-Newtonian power law behavior, viscous dissipation, the Cattaneo-Christov heat flux model, porous medium effects, the Hall effect, and Magnetic field influence. Utilizing similarity transformations, the non-linear PDEs are transformed into non-linear ODEs which are solved numerically using Runge-Kutta-Fehlberg with shooting technique coding in Python. The effects of various thermo-physical and geometrical parameters on the non-dimensional velocity f '(η), transverse velocity h(η), and temperature θ(η) profiles are meticulously analyzed through graphical representations and numerical data. Results demonstrate a significant reduction in fluid velocity f '(η) components and increasing in transverse velocity profile h(η) with increasing magnetic field strength M. Additionally, variations in the thermal relaxation parameter 𝛤 result in substantial changes in the Nusselt number 𝑁𝑢𝑥, indicating enhanced heat transfer rates. Additionally, the study finds that changes in the thermal relaxation parameter 𝛤 impact the Biot number Bi, diminishing its effect. The detailed impacts of other parameters are discussed extensively in the study. These findings provide valuable insights for industrial applications, particularly in developing advanced cooling fluids for automotive and rocket engines. Enhanced heat dissipation and efficient thermal management are crucial for improving engine performance, reliability, and lifespan in these high-demand applications.