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The present study introduces a novel algorithm based on the homotopy analysis method (HAM) to efficiently solve the equation of motion of simply supported transversely and axially loaded double-beam systems. The original HAM was developed for single partial differential equations (PDEs); the current formulation applies to systems of PDEs. The system of PDEs is derived by modeling two prismatic beams interconnected by a nonlinear inner layer as Euler–Bernoulli beams. We employ the Bubnov–Galerkin technique to turn the PDEs’ system into a system of ordinary differential equations that is further solved with the HAM. The flexibility and straightforwardness of the HAM in computing time-dependent components of the system’s transverse deflection and natural frequencies, in conjunction with the observed fast convergence, offer a robust semi-analytical method for analyzing such systems. Finally, the transverse deflection is built through the modal superposition principle. Thanks to a judicious and high-flexibility selection of initial guesses and convergence control parameters, numerical examples confirm that at most six iterations are needed to achieve convergence, and the results are consistent with the selected benchmark cases.