  We continue the research on the asymptotic and preasymptotic decay of
singular numbers for tensor product Hilbert-Sobolev type embeddings in high
dimensions with special emphasis on the influence of the underlying dimension
$d$. The main focus in this paper lies on tensor products involving univariate
Sobolev type spaces with different smoothness. We study the embeddings into
$L_2$ and $H^1$. In other words, we investigate the worst-case approximation
error measured in $L_2$ and $H^1$ when only $n$ linear samples of the function
are available. Recent progress in the field shows that accurate bounds on the
singular numbers are essential for recovery bounds using only function values.
The asymptotic bounds in our setting are known for a long time. In this paper
we contribute the correct asymptotic constant and explicit bounds in the
preasymptotic range for $n$. We complement and improve on several results in
the literature. In addition, we refine the error bounds coming from the setting
where the smoothness vector is moderately increasing, which has been already
studied by Papageorgiou and Wo{\'z}niakowski.
