Product-action wreath products are primitive under the standard hypotheses

Let $B\subseteq {\mathrm{\Delta }}^{\mathrm{\ell }}$ be a block containing distinct points $x$ and $y$, and choose a coordinate $i$ with $x_i\mathrm{\ne }{y}_i$. By [A2], some $h\in H_{x_i}$ moves $y_i$. Let $a\in H^{\mathrm{\ell }}$ act as $h$ in coordinate $i$ and trivially elsewhere. Then $a$ fixes $x$, so $aB\cap B\mathrm{\ne }\mathrm{\varnothing }$ and the block property gives $aB=B$. Hence $y,ay\in B$ are distinct and differ in exactly coordinate $i$.

Fixing the other coordinates, the set of possible entries in coordinate $i$ among points of $B$ is a block for $H$. It contains the two distinct entries from step 1.1, so primitivity of $H$ makes it all of $\mathrm{\Delta }$. Thus $B$ contains the entire $i$-coordinate fibre through $y$.

The stabilizer in $H\wr K$ of any point of ${\mathrm{\Delta }}^{\mathrm{\ell }}$ induces the transitive group $K$ on the coordinates: a coordinate permutation can be followed by coordinatewise elements of the transitive group $H$ to restore the point. Applying these point-stabilizer elements to the fibre in step 2.1 gives a full fibre in every coordinate. Independent coordinate changes then show that $B={\mathrm{\Delta }}^{\mathrm{\ell }}$. Hence every block is a singleton or the whole set, so the product action is primitive.