Cellulose nanocrystals (CNC) are naturally-derived nanostructures of growing importance for the production of composites having attractive mechanical properties, and offer improved sustainability over purely petroleum-based alternatives. Fabrication of CNC composites typically involves extrusion of CNC suspensions and gels in a variety of solvents, in the presence of additives such as polymers and curing agents. However, most studies so far have focused on aqueous CNC gels, yet the behavior of CNC-polymer gels in organic solvents is important to their wider processability. Here, we study the rheological behavior of composite polymer-CNC gels in dimethylformamide, which include additives for both UV and thermal crosslinking. Using rheometry coupled with in-situ infrared spectroscopy, we show that under external shear, CNC-polymer gels display progressive and irreversible failure of the hydrogen bond network that is responsible for their pronounced elastic properties. In the absence of cross-linking additives, the polymer-CNC gels show negligible recovery upon cessation of flow, while the presence of additives allows the gels to recover via van der Waals interactions. By exploring a broad range of shear history and CNC concentrations, we construct master curves for the temporal evolution of the viscoelastic properties of the polymer-CNC gels, illustrating universality of the observed dynamics with respect to gel composition and flow conditions. We therefore find that polymer-CNC composite gels display a number of the distinctive features of colloidal glasses and, strikingly, that their response to the flow conditions encountered during processing can be tuned by chemical additives. These findings have implications for processing of dense CNC-polymer composites in solvent casting, 3D printing, and other manufacturing techniques.