Herein we report the synthesis of new reactive stimuli-responsive polymers by RAFT copolymerization of glycidyl methacrylate and three cyclic N-vinyllactam derivatives. The copolymerization process was thoroughly investigated and the influence of the steric hindrance originating from the monomer structure of cyclic N-vinyllactams on the polymerization process and the properties of obtained copolymers were studied. A series of water-soluble copolymers with variable chemical composition, controlled molecular weight and narrow dispersity (D) were synthesized and their properties are systematically investigated. Experimentally determined cloud points for different copolymers in aqueous solutions indicate shift of lower critical solution temperature (LCST) to lower values with the increase of GMA content in copolymers and increase of the lactam ring size. The obtained reactive stimuli-responsive copolymers can be efficiently used for encapsulation of cellulase in water-in-oil emulsions forming biohybrid nanogels. The enzymes entrapped in nanogels demonstrated significantly improved resistance against harsh store conditions, chaotropic agents, and organic solvents.