Motivated by the long-standing hints of lepton-flavour non-universality in the b → cℓν and b → sℓ+ℓ− 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 mathcal{O} (αs) corrections to the pp → 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 fb−1 of proton-proton collisions at sqrt{s} = 13 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.