Quantum resources promise to surpass classical limits in measurement precision, enabling sensors to reach ultimate sensitivity bounds. In conventional optical gyroscopes using laser probes, the angular velocity sensitivity is fundamentally constrained by the shot-noise limit. Here, we propose a hybrid nonlinear-linear optical gyroscope that integrates both nonlinear and linear interferometric elements to achieve sensitivity enhancements beyond this classical limit. We theoretically analyze the performance of the proposed scheme under various input seeding conditions and evaluate its robustness against realistic experimental imperfections, including photon losses and detection inefficiency for two different intensity detection schemes, i.e., intensity sum and intensity difference. Our results demonstrate that the gyroscope configuration is resilient to inefficient detection; even in the presence of photon losses, the coherent seeded configurations show significant quantum enhancement. Specifically, with 10% photon loss in every optical path and 10% detection inefficiency in intensity-difference measurement, the proposed configuration with a coherent seed input yields 9 dB of quantum enhancement in sensitivity beyond the classical limit when probed with two-mode squeezed states with 13 dB of squeezing. These findings suggest that hybrid nonlinear-linear optical gyroscopes provide a robust pathway for the realization of quantum-enhanced navigation and inertial sensing.