Given a representation $U$ of $S_d$ and $m \in \mathbb{N}$, we can extend the action of $S_d \times \cdots \times S_d$ on $U \otimes \cdots \otimes U = U^{\otimes m}$ to the wreath product $S_d \wr S_m$ by making $S_m$ act on the $m$ factors by place permutation. Let $U^{\widetilde{\otimes m}}$ denote this representation of $S_d \wr S_m$. The representation in the question is then $V^{\widetilde{\otimes n}}$ where $V$ is the natural $d$-dimensional representation of $S_d$.

Conjugacy classes and irreducible representations of wreath products are classified in Chapter 4 of *The representation theory of the symmetric groups* by James and Kerber. In particular any representation induced from a tensor product of irreducibles $U^{\widetilde{\otimes n_i}}$ for subgroups $S_d \wr S_{n_i} \le S_d \wr S_n$, where $S_{n_1} \times \cdots \times S_{n_\ell}$ is a Young subgroup of $S_n$, is irreducible, and all irreducibles are obtained by tensoring these modules with the inflations to $S_d \wr S_n$ of irreducible representations of $S_n$.

Given all this theory, questions like the one above become routine. Over any field $F$ of characteristic zero, $V$ decomposes as $F\oplus S$ where $F$ is the trivial representation and $S = \langle e_i - e_j : 1 \le i < j \le d \rangle$ is a $(d-1)$-dimensional irreducible representation (isomorphic to the Specht module $S^{(n-1,1)}$). Now

$$ V^{\widetilde{\otimes n}} \cong \bigoplus_{m=0}^{n} \bigl( F^{\widetilde{\otimes n-m}} \boxtimes S^{\widetilde{\otimes m}} \bigr)\bigl\uparrow_{S_{n-m} \times S_{m}}^{S_m} $$

where each summand is irreducible. To see this, use basic Clifford theory to show that each summand on the right-hand side occurs in the left-hand side, and finish by counting dimensions:

$$ d^n = \sum_{m=0}^n \binom{n}{m} (d-1)^m. $$

In particular there are summands $F^{\widetilde{\otimes n}}$ and $S^{\widetilde{\otimes n}}$.