  We introduce two scalar leptoquarks, the SU$(2)_L$ isosinglet denoted
$\phi\sim(\mathbf{3}, \mathbf{1}, -1/3)$ and the isotriplet
$\varphi\sim(\mathbf{3}, \mathbf{3}, -1/3)$, to explain observed deviations
from the standard model in semi-leptonic $B$-meson decays. We explore the
regions of parameter space in which this model accommodates the persistent
tensions in the decay observables $R_{D^{(*)}}$, $R_{K^{(*)}}$, and angular
observables in $b\to s \mu\mu$ transitions. Additionally, we exploit the role
of these exotics in existing models for one-loop neutrino mass generation
derived from $\Delta L=2$ effective operators. Introducing the vector-like
quark $\chi \sim (\mathbf{3}, \mathbf{2}, -5/6)$ necessary for lepton-number
violation, we consider the contribution of both leptoquarks to the generation
of radiative neutrino mass. We find that constraints permit simultaneously
accommodating the flavour anomalies while also explaining the relative
smallness of neutrino mass without the need for cancellation between leptoquark
contributions. A characteristic prediction of our model is a rate of
muon--electron conversion in nuclei fixed by the anomalies in $b \to s \mu \mu$
and neutrino mass; the COMET experiment will thus test and potentially falsify
our scenario. The model also predicts signatures that will be tested at the LHC
and Belle II.
