In this paper, we consider a class of nonlinear time-space fractional reaction-diffusion equations by transforming the time-space fractional reaction-diffusion equations into an abstract evolution equations in a fractional Sobolev space. Based on operator semigroup theory, the local uniqueness of mild solutions to the reaction-diffusion equations is obtained under the assumption that nonlinear function is locally Lipschitz continuous. On this basis, a blowup alternative result for unique saturated mild solutions is obtained. We further verify the Mittag-Leffler-Ulam-Hyers stability of the nonlinear time-space fractional reaction-diffusion equations.