We study a time--space nonlocal diffusion equation driven by additive time--space white noise, where the time derivative is the Caputo derivative of order $α\in(0,2)$. The model couples local diffusion with a nonlocal convolution operator generated by a radial probability density, thus incorporating memory effects and long-range spatial interactions. For Dirac initial data, we derive an explicit solution formula in the space of tempered distributions, decomposing the solution into a deterministic part and a stochastic convolution kernel expressed through Mittag--Leffler functions. Our main contribution is a sharp characterization of the existence of mild solutions in terms of $α$, the spatial dimension $N$, and the coefficients of the local and nonlocal diffusion terms. In particular, when the Laplacian term is absent, no mild solution exists, whereas for $λ>0$ the admissible regimes depend critically on $(α,N)$, extending and sharpening the known results for purely local fractional stochastic heat equations. Numerical simulations illustrate the evolution of the mean and variance and emphasize subdiffusive spreading and memory effects.