The polyaddition between dicyclic carbonates and diamines leading to poly(hydroxy urethane)s (PHUs) has emerged as the preferred method for the synthesis of green, non-isocyanate polyurethanes. However, when proposed for use as structural adhesives, the long times for completion of aminolysis of the 5-membered cyclic carbonates under ambient conditions force the use of complementary chemistries to accelerate the curing process. In this work, a system that combines an amino-terminated PHU (NH2-PHU-NH2), an epoxy resin, and a thiol compound was employed to develop high-shear strength PHU-epoxy hybrid adhesives able to cure at room temperature in short times. A NH2-PHU-NH2 prepolymer synthesized by using a sub-stoichiometric quantity of dicyclic carbonates was mixed with a bisphenol A-based epoxy resin for the preparation of the structural adhesive. While this adhesive showed good lap-shear strength and shear resistance under static load and temperature, the curing process was slow. In order to speed up the curing process, a thiol (trimethylolpropane tris(3-mercapto propionate)) was added and its impact on the curing process as well as on the adhesive properties was evaluated. The trifunctional thiol additive allowed for faster curing in the presence of the 1,1,3,3-tetramethylguanidine basic catalyst. Moreover, a combination of NH2-PHU-NH2 and the thiol as curing agents for the epoxy resin resulted in adhesives with superior toughness, without any deterioration of the ultimate lap-shear strength or shear resistance under load and temperature, making these adhesives suitable for high-demand applications in the automotive industry.