In this study, closed-form analytical solutions were developed to predict the effective transverse Young’s modulus, Poisson’s ratios, and mechanical responses of a 3-phase nanocomposite cylindrical model, representing a nanocomposite unit cell, subjected to uniform external pressure. The model is composed of three concentric tubes with distinct generally cylindrical orthotropic properties, representing a carbon nanotube (CNT) surrounded by a matrix and a distinct interphase region in between them. To verify the analytically derived solutions, the unit check was performed at every stage of the formula development and the units for the final solutions were all consistent and verified. For further verification, three different sets of orthotropic materials properties and dimensions were assigned to the constituents and numerical values were obtained for the effective material properties that agreed with our expectations. Furthermore, a uniform external pressure was applied to the outside boundary of the three-phase nanocomposite tube to investigate and verify the mechanical responses (e.g., radial displacements and axial stresses) of the nanocomposite unit cell model. To demonstrate the applicability of the analytical solutions for parametric studies, design analysis, and optimization, one can investigate the variation of the effective transverse Young’s modulus as a function of the constituents’ volume contents and study the stress and strain distributions.