  We study generalisations of conjugacy separability in restricted wreath
products of groups. We provide an effective upper bound for
$\mathcal{C}$-conjugacy separability of a wreath product $A \wr B$ in terms of
the $\mathcal{C}$-conjugacy separability of $A$ and $B$, the growth of
$\mathcal{C}$-cyclic subgroup separability of $B$, and the
$\mathcal{C}$-residual girth of $B.$ As an application, we provide a
characterisation of when $A \wr B$ is $p$-conjugacy separable. We use this
characterisation to the provide for each prime $p$ an example of wreath
products with infinite base group that are $p$-conjugacy separable. We also
provide asymptotic upper bounds for conjugacy separability for wreath products
of nilpotent groups which include the lamplighter groups and provide asymptotic
upper bounds for conjugacy separability of the free metabelian groups. Along
the way, we provide a polynomial upper bound for the shortest conjugator
between two elements of length at most $n$ in a finitely generated nilpotent
group.
