A simultaneous measurement of the three components of the top-quark and top-antiquark polarisation vectors in t-channel single-top-quark production is presented. This analysis is based on data from proton-proton collisions at a centre-of-mass energy of 13 TeV corresponding to an integrated luminosity of 139 fb(-1), collected with the ATLAS detector at the LHC. Selected events contain exactly one isolated electron or muon, large missing transverse momentum and exactly two jets, one being b-tagged. Stringent selection requirements are applied to discriminate t-channel single-top-quark events from the background contributions. The top-quark and top-antiquark polarisation vectors are measured from the distributions of the direction cosines of the charged-lepton momentum in the top-quark rest frame. The three components of the polarisation vector for the selected top-quark event sample are P-x' = 0.01 +/- 0.18, P-y' = -0.029 +/- 0.027, P-z' = 0.91 +/- 0.10 and for the top-antiquark event sample they are P-x' = -0.02 +/- 0.20, P-y' = -0.007 +/- 0.051, P-z' = -0.79 +/- 0.16. Normalised differential cross-sections corrected to a fiducial region at the stable-particle level are presented as a function of the charged-lepton angles for top-quark and top-antiquark events inclusively and separately. These measurements are in agreement with Standard Model predictions. The angular differential cross-sections are used to derive bounds on the complex Wilson coefficient of the dimension-six O-tW operator in the framework of an effective field theory. The obtained bounds are C-tW is an element of[-0.9, 1.4] and C-itW is an element of [-0.8, 0.2], both at 95% confidence level.