Soft actuators have the advantages of compliance and adaptability when working with vulnerable objects, but the deformation shape of the soft actuators is difficult to measure or estimate. Soft sensors made of highly flexible and responsive materials are promising new approaches to the shape estimation of soft actuators, but suffer from highly nonlinear, hysteresis, and time-variant properties. A nonlinear and adaptive state observer is essential for the shape estimation from soft sensors. Current state estimation methods rely on complex nonlinear data-fitting models, and the robustness of the estimation methods is questionable. This study investigates the soft actuator dynamics and the soft sensor model as a stochastic process characterized by the Gaussian Process (GP) model. The unscented Kalman filter (UKF) is applied to the GP model for more reliable variance adjustment during the sequential state estimation process than conventional methods. In addition, a major limitation of the GP model is its computational complexity during online inference. To improve the real-time performance while guaranteeing accuracy, we introduce an edge server to decrease the onboard computational and memory overhead. The experiments showcase a significant improvement in estimation accuracy and real-time performance compared to baseline methods.