  We give a construction of $G_2$ and $Spin(7)$ instantons on exceptional
holonomy manifolds constructed by Bryant and Salamon, by using an ansatz of
spherical symmetry coming from the manifolds being the total spaces of rank-4
vector bundles. In the $G_2$ case, we show that, in the asymptotically conical
model, the connections are asymptotic to Hermitian Yang-Mills connections on
the nearly K\"ahler $S^3\times S^3$.
