In the case $\Omega=\mathbb{R}^n$ we have
\begin{equation}
  L^2(0,T;W^{1,2}(\Omega))\cap W^{1,2}(0,T;W^{-1,2}(\Omega))\hookrightarrow BUC([0,T];X)
\end{equation}
where $X$ is given via real interpolation:
`\begin{equation}
  X=\Big(W^{1,2}(\Omega)),W^{-1,2}(\Omega))\Big)_{1/2,2}=B^0_{2,2}(\Omega).
\end{equation}`
This is basically contained in *Linear and quasilinear parabolic problems I* by H. Amann (Theorem III.4.10.2).
Since $B^0_{2,2}(\Omega)=L^2(\Omega)$ the desired result follows. Now the case of smooth bounded $\Omega$ should follow via extension and restriction.