We aim at insights about how localization of the background field impacts nonlinear quantum vacuum signatures probed by photons in purely magnetic, electric and crossed fields. The starting point of our study are the one-loop results for the Heisenberg-Euler effective Lagrangian and the photon polarization tensor in quantum electrodynamics (QED) evaluated in a uniform constant electromagnetic field. As is well known and often employed, especially in the weak-field limit, within certain restrictions these results also allow for the reliable analysis of vacuum polarization effects in slowly varying background fields. Here, our main interest is in manifestly nonperturbative effects. To this end, we make use of the fact that for the particular case of background field inhomogeneities of Lorentzian shape with $0\le d\le 3$ inhomogeneous directions analytical insights are possible. We study the scaling of conventional nonlinear QED signatures, such as probe-photon polarization flip and probe-photon induced electron-positron pair production, with relevant parameters. Special attention is put on the $d$ dependence of the considered effects.