The current numerical investigation aims to study buoyancy-driven convection of hybrid nanofluids in an annular enclosure formed by two vertical concentric cylinders. In this analysis, hybrid nanofluid containing water and Ag-MgO nanoparticles has been taken as the working medium in the annular domain. The outer wall of the annulus is maintained at lower temperature and adiabatic condition at the horizontal walls of the annulus has been considered. However, along the inner wall, two different thermal conditions are imposed. For Case-I, a linear temperature profile has been considered, and for Case-II, a uniform temperature has been considered. For both the linear and uniform heating, the impact of Rayleigh number, nanoparticle concentration, and different proportions of nanoparticles on fluid flow and thermal transport characteristics in the vertical annulus has been addressed. The numerical simulations are performed for a vast range of parameters to examine fluid flow and thermal transport characteristics in the annular enclosure. The results are represented graphically through flow and thermal contours, and local and average Nusselt numbers. It is noticed that the presence of nanoparticles greatly helps in enhancing the heat removal rate in the annulus. In addition, several numerical computations have been carried out to identify the optimum thermal boundary condition to achieve enhanced heat transport rate.