We study lepton flavour universality violation in SM boson decays in low-scale seesaw models of neutrino mass generation, also addressing other electroweak precision observables. We compute the electroweak next-to-leading order corrections, which turn out to be important - notably in the case of the invisible decay width of the $Z$ boson, for which the corrections can be as large as the current experimental uncertainty. As a well-motivated illustrative study case, we choose a realisation of the Inverse Seesaw mechanism, and discuss the complementary role of lepton flavour conserving, lepton flavour violating and precision observables, both in constraining and in probing such models of neutrino mass generation. Our findings suggest that invisible $Z$ decays are especially important, potentially at the origin of the most stringent constraints for certain regimes of the Inverse Seesaw (while complying with charge lepton flavour violation and other electroweak precision tests). We also discuss the probing power of the considered observables in view of the expected improvement in experimental precision at FCC-ee.