Previously, all-atom molecular dynamics (MD) simulations of a single hydrophobic drug molecule in a series of poly (ethylene oxide-b-caprolactone) (PEO-b-PCL) pseudo-micelles were performed to gain insight into the drug-polymer interactions and drug diffusion. Although some insights into the hydrogen-bonding interactions could be obtained from these simulations, it was not possible to capture the full effect of the interactions on the drug diffusion dynamics. All-atom MD simulations of drug diffusion from stable drug-loaded micelles are prohibitively costly due to the very long timescales associated with drug release from such systems. To reduce the computational cost, in this study, we performed an all-atom MD simulation starting from a disordered structure of a full Cucurbitacin B (CuB)-loaded PEO-b-PCL block copolymer micelle in water. We found that the ensuing drug diffusion out of the micelle is mediated first by the formation of water-PCL bonds and breaking of water-PEO bonds, and then by the formation of water-CuB bonds. The CuB and water dynamics yielded nonlinear sub-diffusive mean-squared displacements, owing to the molecular crowding in the micelle and hydrogen bonding interactions between the water/CuB molecules and polymer chains. Finally, we found that the hydrogen bonding and diffusion dynamics in the pseudo-micelle are not representative of those in the full micelle. The computational approach used in this study is expected to yield molecular-level information that can aid in understanding in-vitro drug release data from nano-sized polymer micelles.