A search for direct pair production of scalar partners of the top quark (top squarks or scalar third-generation up-type leptoquarks) in the all-hadronic $t{\bar{t}}$ plus missing transverse momentum final state is presented. The analysis of 139 $\hbox {fb}^{-1}$ of ${\sqrt{s}=13}$ TeV proton–proton collision data collected using the ATLAS detector at the LHC yields no significant excess over the Standard Model background expectation. To interpret the results, a supersymmetric model is used where the top squark decays via ${\tilde{t}} \rightarrow t^{(*)} {\tilde{\chi }}^0_1$, with $t^{(*)}$ denoting an on-shell (off-shell) top quark and ${\tilde{\chi }}^0_1$ the lightest neutralino. Three specific event selections are optimised for the following scenarios. In the scenario where $m_{{\tilde{t}}}> m_t+m_{{\tilde{\chi }}^0_1}$, top squark masses are excluded in the range 400–1250 GeV for ${\tilde{\chi }}^0_1$ masses below 200 GeV at 95% confidence level. In the situation where $m_{{\tilde{t}}}\sim m_t+m_{{\tilde{\chi }}^0_1}$, top squark masses in the range 300–630 GeV are excluded, while in the case where $m_{{\tilde{t}}}< m_W+m_b+m_{{\tilde{\chi }}^0_1}$ (with $m_{{\tilde{t}}}-m_{{\tilde{\chi }}^0_1}\ge 5$ GeV), considered for the first time in an ATLAS all-hadronic search, top squark masses in the range 300–660 GeV are excluded. Limits are also set for scalar third-generation up-type leptoquarks, excluding leptoquarks with masses below 1240 GeV when considering only leptoquark decays into a top quark and a neutrino.