The computational homogenization approach has been extensively used to determine the constitutive relations of composite materials that would otherwise be analytically intractable. This approach, traditionally developed to determine the constitutive response of continua, has been extended over the last decade to capture the constitutive response of composite structural elements. This study addresses for the first time the computational homogenization of geometrically exact hyperelastic rods with nonuniform warping. The inclusion of cross-sectional distortion (warping) modes at the macroscale makes the formulation suitable for capturing the response near boundaries where Saint-Venant's principle is violated. This is necessary for the simulation of elements in which the influence of cross-sectional distortion boundary conditions spans distances in the order of the element's length, such as squat elements or thin-walled members. The work details the transition between scales, both of which consider finite deformations, by making use of a generalization of the Hill-Mandel condition. Displacement, periodic, and traction boundary conditions are explored. The homogenization process is decoupled to alleviate the high computational cost of the multiscale structural model for nonlinear deformations. The mesoscale simulations are performed offline, and a surrogate hyperelastic potential, which acts as the material constitutive law in the online rod simulations, is calibrated on the basis of the computed homogenized energy, stress resultants, and cross-sectional rigidity. The development of a surrogate for the homogenized response of the rod introduces challenges distinct from those that arise for a homogenized continuum, such as a higher-dimensional strain space and the reduced interpretability of some of the rod's strains. Steps taken to address these difficulties in training data generation, surrogate model selection, and surrogate calibration are addressed. Representative numerical examples are presented to evaluate the efficiency of the surrogate constitutive model and the overall performance of the decoupled two-scale geometrically exact rod against full three-dimensional high-fidelity finite element simulations.