This paper delves into the Terwilliger algebra associated with the ordered Hamming scheme, which extends from the wreath product of one-class association schemes and was initially introduced by Delsarte as a natural expansion of the Hamming schemes. Levstein, Maldonado and Penazzi have shown that the Terwilliger algebra of the Hamming scheme of length $n$ is the $n$-fold symmetric tensor algebra of that of the one-class association scheme. Furthermore, Bhattacharyya, Song and Tanaka have established that the Terwilliger algebra of the wreath product of a one-class association scheme is a direct sum of the ``primary'' subalgebra and commutative subalgebras. This paper extends these findings to encompass both conclusions.