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We present a model of radiative neutrino masses which also resolves anomaliesreported in $B$-meson decays, $R_{D^{(\star)}}$ and $R_{K^{(\star)}}$, as wellas in muon $g-2$ measurement, $\Delta a_\mu$. Neutrino masses arise in themodel through loop diagrams involving TeV-scale leptoquark (LQ) scalars $R_2$and $S_3$. Fits to neutrino oscillation parameters are obtained satisfying allflavor constraints which also explain the anomalies in $R_{D^{(\star)}}$,$R_{K^{(\star)}}$ and $\Delta a_\mu$ within $1\, \sigma$. An isospin-3/2 Higgsquadruplet plays a crucial role in generating neutrino masses; we point outthat the doubly-charged scalar contained therein can be produced in the decaysof the $S_3$ LQ, which enhances its reach to 1.1 (6.2) TeV at $\sqrt s=14$ TeVhigh-luminosity LHC ($\sqrt s=100$ TeV FCC-hh). We also presentflavor-dependent upper limits on the Yukawa couplings of the LQs to the firsttwo family fermions, arising from non-resonant dilepton ($pp \rightarrow \ell^+\ell^-$) processes mediated by $t$-channel LQ exchange, which for 1 TeV LQmass, are found to be in the range $(0.15 - 0.36)$. These limits preclude anyexplanation of $R_{D^{(\star)}}$ through LQ-mediated $B$-meson decays involving$\nu_e$ or $\nu_\mu$ in the final state. We also find that the same Yukawacouplings responsible for the chirally-enhanced contribution to $\Delta a_\mu$give rise to new contributions to the SM Higgs decays to muon and tau pairs,with the modifications to the corresponding branching ratios being at (2-6)%level, which could be tested at future hadron colliders, such as HL-LHC andFCC-hh.

Unified Framework for $B$-Anomalies, Muon $g-2$, and Neutrino Masses