The unscented Kalman filter is an algorithm capable of handling nonlinear scenarios. Uncertainty in process noise covariance may decrease the filter estimation performance or even lead to its divergence. Therefore, it is important to adjust the process noise covariance matrix in real time. In this paper, we developed an adaptive neural unscented Kalman filter to cope with time-varying uncertainties during platform operation. To this end, we devised ProcessNet, a simple yet efficient end-to-end regression network to adaptively estimate the process noise covariance matrix. We focused on the nonlinear inertial sensor and Doppler velocity log fusion problem in the case of autonomous underwater vehicle navigation. Using a real-world recorded dataset from an autonomous underwater vehicle, we demonstrated our filter performance and showed its advantages over other adaptive and non-adaptive nonlinear filters.