A search for pair production of third-generation scalar leptoquarks decaying into a top quark and a $\tau$-lepton is presented. The search is based on a dataset of $pp$ collisions at $\sqrt{s}=13$ TeV recorded with the ATLAS detector during Run 2 of the Large Hadron Collider, corresponding to an integrated luminosity of 139 fb$^{-1}$. Events are selected with one light lepton (electron or muon) and at least one hadronically decaying $\tau$-lepton, or at least two light leptons. In addition, two or more jets, at least one of which must be identified as containing $b$-hadrons, are required. Six final states, defined by the multiplicity and flavour of lepton candidates, are considered in the analysis. Each of them is split into multiple event categories to simultaneously search for the signal and constrain several leading backgrounds. The most sensitive event categories require at least one hadronically decaying $\tau$-lepton candidate and exploit the presence of energetic final state objects, which is characteristic of signal events. No significant excess above the Standard Model expectation is observed in any of the considered event categories, and 95% CL upper limits are set on the production cross section as a function of the leptoquark mass, under different assumptions for the branching fractions into $t\tau$ and $b\nu$. Scalar leptoquarks decaying exclusively into $t\tau$ are excluded up to masses of $1.43$ TeV while, for a branching fraction of 50% into $t\tau$, the lower mass limit is $1.22$ TeV.