  In this paper, we are concerned with regularity of nonlocal stochastic
partial differential equations of parabolic type. By using Companato estimates
and Sobolev embedding theorem, we first show the H\"{o}lder continuity (locally
in the whole state space $\mathbb{R}^d$) for mild solutions of stochastic
nonlocal diffusion equations in the sense that the solutions $u$ belong to the
space $C^{\gamma}(D_T;L^p(\Omega))$ with the optimal H\"{o}lder continuity
index $\gamma$ (which is given explicitly), where $D_T:=[0,T]\times D$ for
$T>0$, and $D\subset\mathbb{R}^d$ being a bounded domain. Then, by utilising
tail estimates, we are able to obtain the estimates of mild solutions in
$L^p(\Omega;C^{\gamma^*}(D_T))$. What's more, we give an explicit formula
between the two index $\gamma$ and $\gamma^*$. Moreover, we prove H\"{o}lder
continuity for mild solutions on bounded domains. Finally, we present a new
criteria to justify H\"{o}lder continuity for the solutions on bounded domains.
The novelty of this paper is that our method are suitable to the case of
time-space white noise.
