This study deals with determining the nonlocal parameter value to achieve a more accurate axial-buckling response of carbon nanoshells conveying nanofluids, analytically. To this end, the four plates/shells' classical theories of Love, Flügge, Donnell, and Sanders are generalized using Eringen's nonlocal elasticity theory. By combining these theories in cylindrical coordinates, a modified motion equation is presented to investigate the buckling behavior of the nanofluid-nanostructure-interaction problem. Herein, in addition to the small-scale effect of the structure and the passing fluid on the critical buckling strain, the effects of nanoflow velocity, the fluid density (nano-liquid/nano-gas), half-wave numbers, aspect ratio, and the nanoshell flexural-rigidity are discussed. The analytical approach is used for discretizing and solving the obtained relations to study the mentioned cases.