  Motivated by the long-standing hints of lepton-flavour non-universality in
the $b \to c \ell \nu$ and $b \to s \ell^+ \ell^-$ channels, we study Drell-Yan
ditau production at the Large Hadron Collider (LHC). In the context of models
with third-generation gauge vector leptoquarks (LQs), we calculate the complete
${\cal O} (\alpha_s)$ corrections to the $pp \to \tau^+ \tau^-$ process,
achieving next-to-leading order (NLO) plus parton shower (NLO$+$PS) accuracy
using the POWHEG method. We provide a dedicated Monte Carlo code that evaluates
the NLO QCD corrections on-the-fly in the event generation and use it to study
the numerical impact of NLO$+$PS corrections on the kinematic distributions
that enter the existing experimental searches for non-resonant ditau final
states. Based on our phenomenological analysis we derive NLO accurate
constraints on the masses and couplings of third-generation gauge vector LQs
using the latest LHC ditau search results corresponding to an integrated
luminosity of around $140 \, {\rm fb}^{-1}$ of proton-proton collisions at
$\sqrt{s} = 13 \, {\rm TeV}$. The presented NLO$+$PS generator allows for an
improved signal modelling, making it an essential tool for future ATLAS and CMS
searches for vector LQs in $\tau^+ \tau^-$ final states at LHC Run III and
beyond.
