  In this work, a time-fractional nonlocal diffusion equation is considered.
Based on the $L2$-$1_{\sigma}$ scheme on a graded mesh in time and the standard
finite element method (FEM) in space, the fully-discrete $L2$-$1_{\sigma}$
finite element method is investigated for a time-fractional nonlocal diffusion
problem. We prove the existence and uniqueness of fully-discrete solution. The
$\alpha$-robust error bounds are derived, i.e. bounds remain valid as $\alpha$
$\rightarrow {1}^{-},$ where $\alpha \ \in (0,1)$ is the order of a temporal
fractional derivative. The numerical experiments are presented to justify the
theoretical findings.
