[abridged] 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 (alpha ~ -0.6 to -0.7, where F_nu propto nu^alpha). This jet emission is tightly and linearly correlated with the X-ray flux; L_OIR,jet propto L_X^(0.98 +- 0.08) 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 (alpha = -0.73). Ostensibly the compact, synchrotron jet could therefore account for ~ 100 % of the X-ray flux at low luminosities in the hard state. At the same time, (i) an excess is seen over the power law 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 e-4 - 2 e-3) L_Edd 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.
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