The purpose of the current study is to investigate the non-Newtonian unsteady Williamson fluid on a stretching/shrinking surface along with thermophoresis and Brownian effects. Basically, the model consists of a time-dependent magnetic field. The fluid is considered to be electrically conducting due to the effect of the external magnetic field. The values of magnetic Reynolds number are so small that the induced magnetic field is assumed to be negligible. In the concentration equation, the effects of Brownian motion and thermophoresis are discussed. Employing the similarity transformations, the governing nonlinear Partial Differential Equations (PDEs) are converted into the Ordinary Differential Equations (ODEs). The resulting ODEs are solved with the combined effects of the Successive Over Relaxation (SOR) method and Finite Difference Method (FDM). The impact of all the including parameters such as suction parameter, injection parameter, stretching/shrinking parameter, the ratio of viscosity, local Weissenberg number, unsteadiness parameter, Eckert number, Prandtl number, Lewis number, Nusselt number, Brownian motion parameter, shear stress, heat transfer rate, and mass transfer rate are analyzed using graphs and tables. Results show that the values of fluid velocity are better for S=8, −S=0, λ=0.3, β*=0.9, Wi=0.3, and Aa=2.0. It is also depicted from the results that the values of boundary layer thickness are better for S=0, −S=−8, λ=0.3,  β*=0.1, Wi=1.5, and Aa=0.25. From the above numeric results, it is concluded that the fluid velocity is reduced and the thermal boundary layer thickness is enhanced by the enhancement of the stretching parameter.