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.