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Motivated by the long-standing hints of lepton-flavour non-universality in the $b \to c \ell ν$ 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 ${\cal O} (α_s)$ corrections to the $pp \to τ^+ τ^-$ 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 $τ^+ τ^-$ final states at LHC Run III and beyond.