Context. The physics and chemistry of molecular gas change with the dynamical evolution of the newly formed protostars. The details of this evolution are not yet fully understood. Aims. Our goal is to identify chemical diagnostics to determine the physical conditions in protostellar envelopes and to help establish the development of matter during the formation of a low-mass star. In particular, we aim to investigate a possible variation of sulphur depletion during the star formation process at the scale of the cold envelope. Methods. We used observations obtained with the Yebes-40m and IRAM-30 m telescopes to estimate column densities of CS, C2S, C3S, OCS, HCS+, SO, H2CS, SO2, NS, NS+, H2S, HSCN, HNCS, CH3SH, C18O, C17O, CH3OH, H2CO, and their isotopologues in the Class 0 objects HH 212 and NGC 1333 IRAS 4A, using both local thermodynamic equilibrium (LTE) and non-LTE calculations. We used a neural emulator of the chemical code Nautilus to investigate the physical (density, gas temperature, cosmic-ray ionisation rate, and chemical time) and chemical (sulphur elemental abundance) conditions driving the evolution of the chemical composition of the cold envelope. Results. A comparison of the abundances of these species with those of the Class 0 object B 335 reveals a differentiated chemistry. While sulphur-bearing species containing carbon chains (C2S, C3S) are between three and seven times more abundant in B 335 than in the other two objects, suggesting an earlier chemistry in this protostar, sulphur oxides and nitrogen-bearing species are approximately three times more abundant in NGC 1333 IRAS 4A. Comparing with similar studies in pre-stellar and protostellar cores, we derive an increase of about two orders of magnitude in the SO/CS and SO2/C2S ratios and a potential decrease in the HCS+/CS ratio of a factor of ten during the transition from the pre-stellar phase to the Class 0 phase. Our chemical modelling shows that while the chemistry of HH 212 and NGC 1333 IRAS 4A is well reproduced assuming a gas temperature of ∼25 K, we find significant differences in their average density and cosmic-ray ionisation rate. We estimate a sulphur depletion of a factor of ∼100 in their protostellar envelopes, which is similar to that measured in starless cores. Conclusions. Sulphur compounds are good evolutionary tracers of the pre- to protostellar phase transition, with oxygen-bearing species being more abundant than carbon-bearing species in evolved sources. These chemical differences are the consequence of envelope heating by the forming star and the environmental conditions. We do not detect any significant change in sulphur depletion in the cold envelope during the Class 0 phase.