In this paper, we study the robust beamforming design for a simultaneous wireless information and power transfer (SWIPT) enabled system, with cooperative non-orthogonal multiple access (NOMA) protocol applied. A novel cooperative NOMA scheme is proposed, where the strong user with better channel conditions adopts power splitting (PS) scheme and acts as an energy-harvesting relay to transmit information to the weak user. The presence of channel uncertainties is considered and incorporated in our formulations to improve the design robustness and communication reliability. Specifically, only imperfect channel state information is assumed to be available at the base station (BS), due to the reason that the BS is far away from both users and suffers serious feedback delay. To comprehensively address the channel uncertainties, two major design criteria are adopted, which are the outage-based constraint design and the worst-case-based optimization. Then, our aim is to maximize the strong user's data rate, by optimally designing the robust transmit beamforming and PS ratio, while guaranteeing the correct decoding of the weak user. With two different channel uncertainty models respectively incorporated, the proposed formulations yield to challenging nonconvex optimization problems. For the outage-based constrained optimization, we first conservatively approximate the probabilistic constraints with the Bernstein-type inequalities, which are then globally solved by 2-D exhaustive search. To further reduce the complexity, an efficient low-complexity algorithm is then proposed with the aid of successive convex approximation (SCA). For the worst-case-based scenario, we first apply the semidefinite relaxation method to relax the quadratic terms and prove the rank-one optimality. Then the nonconvex max-min optimization problem is readily transformed into convex approximations based on S-procedure and SCA. Simulation results show that for both channel uncertainty models, the proposed algorithms can converge within a few iterations, and the proposed SWIPT-enabled robust cooperative NOMA system achieves better system performance than the existing protocols.