Consider percolation on $T\times \mathbb{Z}^d$, the product of a regular tree of degree $k\geq 3$ with the hypercubic lattice $\mathbb{Z}^d$. It is known that this graph has $0<p_c<p_u<1$, so that there are non-trivial regimes in which percolation has $0$, $\infty$, and $1$ infinite clusters a.s., and it was proven by Schonmann (1999) that there are infinitely many infinite clusters a.s. at the uniqueness threshold $p=p_u$. We strengthen this result by showing that the Hausdorff dimension of the set of accumulation points of each infinite cluster in the boundary of the tree has a jump discontinuity from at most $1/2$ to $1$ at the uniqueness threshold $p_u$. We also prove that various other critical thresholds including the $L^2$ boundedness threshold $p_{2\to 2}$ coincide with $p_u$ for such products, which are the first nonamenable examples proven to have this property. All our results apply more generally to products of trees with arbitrary infinite amenable Cayley graphs and to the lamplighter on the tree.