In this article, a novel robust predefined-time control approach is proposed for a class of bimanual teleoperation systems (BTSs) with system parametric and nonparametric uncertainties. First, an innovative gesture-based teleoperation system configuration is constructed by applying an openpose algorithm, and unmarked two-arm posture recognition is realized simultaneously, thereby the operator can operate the remote two-slave robots in a more natural and friendly way. Then, a new predefined-time sliding mode control (PTSMC) scheme is designed by developing a new nonsingular predefined-time sliding mode (NPTSM) surface without any switching action and applying an online estimation method based on time delay estimation (TDE). By blending the advantages of the TDE and the added nonlinear variable exponent robust terms, predefined time tracking can be achieved. Moreover, in the presence of system uncertainties, a new robust two-layer model-free adaptive sliding mode (ASM) force/torque estimator is designed to obtain the force/torque information in a predefined time. Rigorous theoretical analyses are given by applying Lyapunov functions. Experimental studies using robotic manipulators are conducted to verify the effectiveness of the proposed control algorithms.