The integration of wireless-powered communication and backscatter communication (BackCom) is envisioned as a promising communication paradigm for the future Internet of Things (IoT), where resource allocation (RA) as a critical technique can improve achievable capacity by dynamically adjusting system parameters (e.g., transmit power, beamforming vectors). However, the existing RA algorithms consider the ideal cases of perfect channel state information and linear energy harvesting (EH), which will deteriorate system performance. To fill this gap, in this paper, we investigate a robust RA problem for a wireless-powered BackCom system with non-orthogonal multiple access (NOMA) under the consideration of practical channel uncertainties and nonlinear EH models. The robust RA problem with Euclidean spherical uncertainties is formulated by optimizing the transmit beamforming vector of the access point (AP), the transmit power of each IoT device, and time allocation factors, where the maximum power constraint of the AP, the time frame constraint, the minimum EH constraint, and the quality of service constraint of each user are considered simultaneously. Subsequently, according to the block coordinate descent iterative method, an iteration-based robust RA algorithm is proposed by using the successive convex approximation, S-procedure, semidefinite relaxation, and Gaussian randomization methods. Simulation results prove that our proposed algorithm can drastically enhance system robustness compared to the non-robust algorithm and several benchmark algorithms.