We investigated the Ree–Eyring fluid flow between two stretchable spinning disks with various stretching rates, along with the thermal enhancement feature of dual-walled carbon nanotubes such as SWCNTs and MWCNTs–water nanoliquids. The influence of thermophoresis and Brownian motion and microorganisms was also investigated properly. Appropriate transformations were applied to change the highly coupled non-linear system of partial differential equations to coupled ordinary differential equations associated with convective boundary conditions, which were then obtained analytically by utilizing HAM. The effects of significant parameters on the distribution function of velocity, temperature, microorganism motile density and concentration have been clarified via detailed sketches. Clearly, we found that the characteristics of the Eckert number enhanced the temperature profile in the two different nanoliquids. The thermal boundary layer thickness was enhanced via frictional heating when the Eckert number significantly increased in an augmentation of energy distribution due to double-spinning disks. Finally, the fluid temperature hikes were a little less significant in single-walled carbon nanotubes (SWCNTs) than multi-walled carbon nanotubes (MWCNTs)–water nanoliquid with a larger Eckart number, EC. Moreover, we noticed that the thickness of the microorganism boundary layer decreased when we elevated Weissenberg number We and rotation parameter G. Thus, multi-walled carbon nanotubes (MWCNTs)–water nanoliquid showed a slightly greater decrease than single-walled carbon nanotubes (SWCNTs)–water nanoliquid.