The black hole X-ray binary XTE J1550-564 was monitored extensively at X-ray, optical and infrared wavelengths throughout its outburst in 2000. We show that it is possible to separate the optical/near-infrared (OIR) jet emission from the OIR disc emission. Focussing on the jet component, we find that as the source fades in the X-ray hard state, the OIR jet emission has a spectral index consistent with optically thin synchrotron emission (α ~ -0.6 to -0.7, where Fν ~ να). This jet emission is tightly and linearly correlated with the X-ray flux; suggesting a common origin. This is supported by the OIR, X-ray and OIR to X-ray spectral indices being consistent with a single power law (α = -0.73). Ostensibly the compact synchrotron jet could therefore account for ~100per cent of the X-ray flux at low luminosities in the hard state. At the same time, (i) an excess is seen over the exponential decay of the X-ray flux at the point in which the jet would start to dominate, (ii) the X-ray spectrum slightly softens, which seems to be due to a high-energy cut-off or break shifting to a lower energy and (iii) the X-ray rms variability increases. This may be the strongest evidence to date of synchrotron emission from the compact, steady jet dominating the X-ray flux of an X-ray binary. For XTE J1550-564, this is likely to occur within the luminosity range ~(2 × 10-4-2 × 10-3) LEdd on the hard-state decline of this outburst. However, on the hard-state rise of the outburst and initially on the hard-state decline, the synchrotron jet can only provide a small fraction (~ a few per cent) of the X-ray flux. Both thermal Comptonization and the synchrotron jet can therefore produce the hard X-ray power law in accreting black holes. In addition, we report a phenomenological change in the OIR spectral index of the compact jet from possibly a thermal distribution of particles to one typical of optically thin synchrotron emission, as the jet increases in energy over these ~20d. Once the steady jet is fully formed and the infrared and X-ray fluxes are linearly correlated, the spectral index does not vary (maintaining α = -0.7) while the luminosity decreases by a factor of 10. These quantitative results provide unique insights into the physics of the relativistic jet acceleration process.