  In this article we study how decompositions of a quasi-lattice ordered group
$(G,P)$ relate to decompositions of the Nica-Toeplitz algebra
$\mathcal{NT}_\mathbf{X}$ and Cuntz-Nica-Pimsner algebra
$\mathcal{NO}_\mathbf{X}$ of a compactly aligned product system $\mathbf{X}$
over $P$. In particular, we are interested in the situation where $(G,P)$ may
be realised as the semidirect product of quasi-lattice ordered groups. Our
results generalise Deaconu's work on iterated Toeplitz and Cuntz-Pimsner
algebras - we show that the Nica-Toeplitz algebra and Cuntz-Nica-Pimsner
algebra of a compactly aligned product system over $\mathbb{N}^k$ may be
realised as $k$-times iterated Toeplitz and Cuntz-Pimsner algebras
respectively.
