Depletion-driven assembly has been widely studied for micron-sized colloids, but questions remain at the nanoscale where the governing physics are impacted by the stabilizing surface ligands or wrapping polymers, whose length scales are on the same order as those of the colloidal core and the depletant. Here, we probe how wrapping colloidal tin-doped indium oxide nanocrystals with polymers affects their depletion-induced interactions and assembly in solutions of polyethylene glycol. Copolymers of polyacrylic acid grafted with polyethylene oxide provide nanocrystal wrappings with different effective polymer graft densities and molecular weights. (Ultra) small angle X-ray scattering, coarse-grained molecular dynamics simulation, and molecular thermodynamic theory were combined to analyze how depletant size and polymer wrapping characteristics affect depletion interactions, structure, and phase behavior. The results show how depletant molecular weight, as well as surface density and molecular weight of polymer grafts, set thresholds for assembly. These signatures are unique to depletion-driven assembly of nanoscale colloids and mirror phase behaviors of grafted nanoparticle--polymer composites. Optical and rheological responses of depletion-driven assemblies of nanocrystals with different polymer shell architectures were probed to learn how their structural differences impact properties. We discuss how these handles for depletion-driven assembly at the nanoscale may provide fresh opportunities for designing responsive depletion interactions and dynamically reconfigurable materials.