We present BALRoGO: Bayesian Astrometric Likelihood Recovery of Galactic Objects, a public code to measure the centres, effective radii, and bulk proper motions of Milky Way (MW) globular clusters and Local Group dwarf spheroidals, whose data are mixed with MW field stars. Our approach presents innovative methods such as surface density fits allowing for strong interloper contamination and proper motion fits using a Pearson VII distribution for interlopers, instead of classic Gaussian-mixture recipes. We also use non-parametric approaches to represent the colour–magnitude diagram of such stellar systems based in their membership probabilities, previously derived from surface density and proper motion fits. The robustness of our method is verified by comparing its results with previous estimates from the literature as well as by testing it on mock data from N-body simulations. We applied BALRoGO to Gaia EDR3 data for over 100 MW globular clusters and nine Local Group dwarf spheroidals, and we provide positions, effective radii, and bulk proper motions. Finally, we make our algorithm available as an open source software.