  Convergence of stochastic integrals driven by Wiener processes $W_n$, with
$W_n \to W$ almost surely in $C_t$, is crucial in analyzing SPDEs. Our focus is
on the convergence of the form $\int_0^T V_n\, \mathrm{d} W_n \to \int_0^T V\,
\mathrm{d} W$, where $\{V_n\}$ is bounded in $L^p(\Omega \times [0,T];X)$ for a
Banach space $X$ and some finite $p > 2$. This is challenging when $V_n$
converges to $V$ weakly in the temporal variable. We supply convergence results
to handle stochastic integral limits when strong temporal convergence is
lacking. A key tool is a uniform mean $L^1$ time translation estimate on $V_n$,
an estimate that is easily verified in many SPDEs. However, this estimate alone
does not guarantee strong compactness of $(\omega,t)\mapsto V_n(\omega,t)$. Our
findings, especially pertinent to equations exhibiting singular behavior, are
substantiated by establishing several stability results for stochastic
transport equations and conservation laws.
