Suppose $\{u_{n}\}_{n}$ is a sequence of smooth functions in space and time such that $u_{n} \to u$ in $L^{\infty}(0,T; L^{\infty}(\Omega))$, where $\Omega \subset \mathbb{R}$ is open, bounded.
Question: Can we say anything about the regularity of $\partial_{t}u$? Does it exist at all?
Of course not. There is no need to consider such fancy spaces. Take $u_n(t, x)=f_n(t)$. Your assumption implies that $f_n\to f$ uniformly, where $f=f(t)$. You cannot conclude anything on $f'$, as you already know.