Consider a uniformly chosen element $X_n$ of the $n$-fold wreath product $\Gamma_n = G \wr G \wr \dots \wr G$, where $G$ is a finite permutation group acting transitively on some set of size $s$. The eigenvalues of $X_n$ in the natural $s^n$-dimensional permutation representation are investigated by considering the random measure $\Xi_n$ on the unit circle that assigns mass $1$ to each eigenvalue. It is shown that if $f$ is a trigonometric polynomial, then $\lim_{n \rightarrow \infty} P\{\int f d\Xi_n \ne s^n \int f d\lambda\}=0$, where $\lambda$ is normalised Lebesgue measure on the unit circle. In particular, $s^{-n} \Xi_n$ converges weakly in probability to $\lambda$ as $n \rightarrow \infty$. For a large class of test functions $f$ with non-terminating Fourier expansions, it is shown that there exists a constant $c$ and a non-zero random variable $W$ (both depending on $f$) such that $c^{-n} \int f d\Xi_n$ converges in distribution as $n \rightarrow \infty$ to $W$.

These results have applications to Sylow $p$-groups of symmetric groups and autmorphism groups of regular rooted trees.