Given a smooth globally hyperbolic (3+1)-dimensional spacetime (M,g) satisfying the Einstein vacuum equations (possibly with cosmological constant) and an inextendible timelike geodesic C, we constructed in Part I a family of metrics g_\epsilon on the complement M_\epsilon\subset M of an ϵ-neighborhood of C with the following behavior: away from C one has g_\epsilon\to g as \epsilon\to 0, while the \epsilon^{-1}-rescaling of g_\epsilon around every point of C tends to a fixed subextremal Kerr metric; and gϵ solves the Einstein vacuum equation modulo \mathcal{O}(\epsilon^\infty) errors. The ultimate goal, achieved in Part III, is to correct g_\epsilon to a true solution on any fixed precompact subset of M by addition of a size \mathcal{O}(\epsilon^\infty) metric perturbation which needs to satisfy a quasilinear wave equation (the Einstein vacuum equations in a suitable gauge). The present paper lays the necessary analytical foundations. We develop a framework for proving estimates for solutions of (tensorial) wave equations on (M_\epsilon,g_\epsilon) which, on a suitable scale of Sobolev spaces, are uniform on ϵ-independent precompact subsets of the original spacetime M. These estimates are proved by combining two ingredients: the spectral theory for the corresponding wave equation on Kerr; and uniform microlocal estimates governing the propagation of regularity through the small black hole, including radial point estimates reminiscent of diffraction by conic singularities and long-time estimates near perturbations of normally hyperbolic trapped sets. As an illustration of this framework, we construct solutions of a toy nonlinear scalar wave equation on (M_\epsilon,g_\epsilon) for uniform timescales and with full control in all asymptotic regimes as \epsilon\to 0.
