A time-fractional singularly perturbed reaction-diffusion problem is considered with Dirichlet initial and boundary conditions. Bounds on the solution and its derivatives are proved via a solution decomposition and a maximum principle; these bounds show that the solution has a weak singularity at the initial time and also has layers (caused by the small parameter) at the boundaries of the spatial domain. We apply the L1 discretisation to fractional derivative on a graded temporal mesh, together with a standard finite element method for the spatial derivatives on a Shishkin spatial mesh. Using our bounds on the derivative of the solution, local in time error estimates, which are uniform in the singular perturbation parameter, are derived.