Fluid viscosity is considered as constant in several boundary layer analyses, but this fluid property can change remarkably when the temperature difference exists in the boundary layer. The Prandtl number and Schmidt number can also change significantly as the fluid viscosity changes depending on temperature. Therefore, this framework is exploring the consequences of varying viscosity and varying Prandtl number on Falkner–Skan flow of Williamson nanofluid over a wedge, plate and stagnation point. The Buongiorno nanofluid model has been employed to manifest the fluid transport properties of the Williamson nanofluid. Similarity transformations are utilized to transform the governing equations into ordinary differential equation and solved numerically using Runge–Kutta (RK) Fehlberg method. Williamson fluid velocity, temperature, concentration, skin friction factor, rate of heat transfer and rate of mass transfer are investigated with emerging parameters, and the outcomes are presented graphically. Computed results manifest that the Williamson nanofluid expresses the opposite nature in velocity and temperature for higher values of Weissenberg number parameter. Positive values of variable viscosity parameter diminish the significance of variable Prandtl number and variable Schmidt number in the boundary layer. Furthermore, it is noticed that the Williamson nanofluid temperature is higher over a plate compared with wedge and stagnation point cases. © 2020, Springer Nature Switzerland AG.