  We study the Ricci flow starting at an SU(2) cohomogeneity-1 metric $g_{0}$
on $\mathbb{R}^{4}$ with monotone warping coefficients and whose restriction to
any hypersphere is a Berger metric. If $g_{0}$ has bounded Hopf-fiber,
curvature controlled by the size of the orbits and opens faster than a
paraboloid in the directions orthogonal to the Hopf-fiber, then the flow
converges to the Taub-NUT metric $g_{\mathsf{TNUT}}$ in the Cheeger-Gromov
sense in infinite time. We also classify the long-time behaviour when $g_{0}$
is asymptotically flat. In order to identify infinite-time singularity models
we obtain a uniqueness result for $g_{\mathsf{TNUT}}$.
