We report on quasi-simultaneous observations from radio to X-ray frequencies of the neutron star X-ray binary Aql X-1 over accretion state transitions during its 2016 outburst. All the observations show radio to millimetre spectra consistent with emission from a jet, with a spectral break from optically thick to optically thin synchrotron emission that decreases from ~100 GHz to <5.5 GHz during the transition from a hard to a soft accretion state. The 5.5 GHz radio flux density as the source reaches the soft state, 0.82 ± 0.03 mJy, is the highest recorded to date for this source. During the decay of the outburst, the jet spectral break is detected again at a frequency of ~30–100 GHz. The flux density is 0.75 ± 0.03 mJy at 97.5 GHz at this stage. This is the first time that a change in the frequency of the jet break of a neutron star X-ray binary has been measured, indicating that the processes at play in black holes are also present in neutron stars, supporting the idea that the internal properties of the jet rely most critically on the conditions of the accretion disc and corona around the compact object, rather than the black hole mass or spin or the neutron star surface or magnetic field.