A novel near-field integrated sensing and communications framework for secure unmanned aerial vehicle (UAV) networks with high time efficiency is proposed. A ground base station (GBS) with large aperture size communicates with one communication UAV (C-UAV) under the existence of one eavesdropping UAV (E-UAV), where the artificial noise (AN) is employed for both jamming and sensing purpose. Given that the E-UAV's motion model is unknown at the GBS, we first propose a near-field localization and trajectory tracking scheme. Specifically, exploiting the variant Doppler shift observations over the spatial domain in the near field, the E-UAV's three-dimensional (3D) velocities are estimated from echo signals. To provide the timely correction of location prediction errors, the extended Kalman filter (EKF) is adopted to fuse the predicted states and the measured ones. Subsequently, based on the real-time predicated location of the E-UAV, we further propose a joint GBS beamforming and C-UAV trajectory design scheme for maximizing the instantaneous secrecy rate, while guaranteeing the sensing accuracy constraint. To solve the resultant non-convex problem, an alternating optimization approach is developed, where the near-field GBS beamforming and the C-UAV trajectory design subproblems are iteratively solved by exploiting the successive convex approximation method. Finally, our numerical results unveil that: 1) the E-UAV's 3D velocities and location can be accurately estimated in real time with our proposed framework by exploiting the near-field spherical wave propagation; and 2) the proposed framework achieves superior secrecy rate compared to benchmark schemes and closely approaches the performance when the E-UAV trajectory is perfectly known.