Forget the indices $t$, one is interested in the distribution of $Y=AX+BU$ conditionally on $X$, where the distribution of $X$ is irrelevant and $U$ is independent on $X$ and with centered normal distribution with variance-covariance $Q$. Thus the distribution of $BU$ is centered normal with variance-covariance matrix $C=BQB^*$ and the conditional distribution of $Y$ conditionally on $[X=x]$ is normal with mean $Ax$ and variance-covariance matrix $C$.
Edit The conditional distribution of $Y$ conditionally on $[X=x]$ has a density if and only if it is full-dimensional normal, which happens if and only if $C$ has rank $n$, that is, if and only if $Q$ and $B$ both have rank $n$. Otherwise, for every $x$ in $\mathbb R^n$, the conditional distribution of $Y$ conditionally on $[X=x]$ is concentrated on a hyperplane of $\mathbb R^n$, thus it has no density with respect to the Lebesgue measure on $\mathbb R^n$.