Polymer vesicles, also named polymersomes, are valuable candidates for drug delivery and micro or nano-reactors applications. As far as drug delivery is concerned, the shape of the carrier is believed to have a strong influence on the biodistribution and cell internalization. Polymersomes can be submitted to an osmotic imbalance when injected in physiological media leading to morphological changes. To understand these osmotic stress-induced variations in membrane properties and shapes, several nano-vesicles made of the graft polymer poly(dimethylsiloxane)-g-poly(ethylene oxide) (PDMS-g-PEO) or the triblock copolymer PEO-b-PDMS-b-PEO were osmotically stressed and observed by light scattering, neutron scattering (SANS) and cryo-transmission electron microscopy (cryo-TEM). Hypotonic shock leads to a swelling of the vesicles, comparable to optically observable giant polymersomes, and hypertonic shock leads to collapsed structures such as stomatocytes and original nested vesicles the latter being only observed for diblocks. Complementary SANS and cryo-TEM experiments are shown to be in quantitative agreement and highlight the importance of the membrane structure on the behavior of these nano-polymersomes under hypertonic conditions as the final morphology reached depends whether or not the copolymers assemble into a bilayer. The vesicle radius and membrane curvature are also shown to be critical parameters for such transformations: the shape evolution trajectory agrees with theoretical models only for large enough vesicle radii above a threshold value around four times the membrane thickness.