This is indeed handled in Section 4.2 of James and Kerber's book on Representations of the Symmetric group.

We also considered this problem in Section 2 our paper describing a computer algorithm for computing conjugacy class representatives in permutation groups.

J. Cannon and D. Holt, Computing conjugacy class representatives in permutation groups,
J. Algebra 300 (2006), 213--222.

I can cut and paste the main bit of theory you need. The problem easily reduces to finding those classes that map onto a specific element of $S_n$.

Let $A$ be any group, let $P$ be a permutation group acting
on the set $\{ 1\ldots d\}$, and let $W := A \wr P$.
Then $W$ is a semidirect product of $A^d$ by $P$, and elements
of $W$ have the form $(g,x)$ with $g \in P$, $x \in A^d$, and
$x = (x_1,\ldots,x_d)$ with $x_i \in A$. In general, for $x \in A^d$,
we denote the $i$-th component of $x$ by $x_i$.
The action of $P$ on $A^d$ in $W$ is given by:
$$(g,1)^{-1}(1,(x_1,\ldots,x_d))(g,1) = (1,(x_{1^{g^{-1}}},\ldots,
x_{d^{g^{-1}}})).$$

Hence, for $g \in P$, $x,z\in  A^d$, we have
\begin{equation}\label{conjeqn}
(1,z)^{-1}(g,x)(1,z) = (g,y), \ \mathrm{where}\ 
y_i = z^{-1}_{i^{g^{-1}}}x_iz_i \ \mathrm{for}\  1 \le i \le d.
\end{equation}

$\textbf{Theorem}$   With the above notation,
let $x^{(1)},\ldots,x^{(k)}$ be representatives of the conjugacy classes of $A$.
Fix an element $g \in P$, and let $r_1,r_2,\ldots,r_s$ be representatives of
the cycles of $g$ in its action on $\{ 1\ldots d\}$. Then
$$
R := \{\,(g,x) \mid  x_i = 1 \ \mathrm{for}\  i \not\in \{r_1,\ldots,r_s\},\,
           x_i \in \{x^{(1)},\ldots,x^{(k)}\}\ \mathrm{for}\
i \in \{r_1,\ldots,r_s\}\,\}
$$
is a set of representatives of the $A^d$-classes of elements of $W$
of the form $(g,x)$ for $x \in A^d$.