  In this article, we study the asymptotic behaviour of conjugacy separability
for wreath products of abelian groups. We fully characterise the asymptotic
class in the case of lamplighter groups and give exponential upper and lower
bounds for generalised lamplighter groups. In the case where the base group is
infinite, we give superexponential lower and upper bounds. We apply our results
to obtain lower bounds for conjugacy depth functions of various wreath products
of groups where the acting group is not abelian.
