We provide a systematic assessment of the order-α⁵ nuclear contributions to the Lamb shift of muonic deuterium, including the accompanying radiative corrections due to vacuum polarisation, up to next-to-next-to-next-to-leading order (N3LO) within the pionless effective field theory (π̸EFT). We also evaluate higher-order corrections due to the single-nucleon structure, which are expected to be the most important corrections beyond N3LO. We find a correlation between the deuteron charge and Friar radii, which can be useful to judge the quality of charge form factor parametrisations. We refine the theoretical description of the 2γ-exchange contribution, especially in the elastic contribution and the radiative corrections, ameliorating the original discrepancy between theory and experiment in the size of 2γ-exchange effects. Based on the experimental Lamb shift of muonic deuterium, we obtain the deuteron charge radius, r_d(μD) = 2.12763(13)_exp(77)_theory fm, which is consistent with (but less precise than) the value obtained by combining the H-D isotope shift with the muonic-hydrogen Lamb shift. The theory uncertainty is evaluated using a Bayesian procedure and is dominated by the truncation of the π̸EFT series.