  We prove uniqueness of representations of Nica-Toeplitz algebras associated
to product systems of $C^*$-correspondences over right LCM semigroups by
applying our previous abstract uniqueness results developed for
$C^*$-precategories. Our results provide an interpretation of conditions
identified in work of Fowler and Fowler-Raeburn, and apply also to their
crossed product twisted by a product system, in the new context of right LCM
semigroups, as well as to a new, Doplicher-Roberts type $C^*$-algebra
associated to the Nica-Toeplitz algebra. As a derived construction we develop
Nica-Toeplitz crossed products by actions with completely positive maps. This
provides a unified framework for Nica-Toeplitz semigroup crossed products by
endomorphisms and by transfer operators. We illustrate these two classes of
examples with semigroup $C^*$-algebras of right and left semidirect products.
