We present low-temperature magnetoresistance measurements on lithographically defined bismuth wires. The phase-coherence time and the spin-orbit scattering time are obtained by analysis of weak antilocalization, with values for the phase-coherence time supported by analysis of the universal conductance fluctuations present in the wires. We find that the phase-coherence time is dominated by electron-phonon scattering above approximate to 2 K and saturates below that temperature, with saturation delayed to a lower temperature in wider wires. The spin-orbit scattering time shows a weak temperature dependence above 2 K, and also shows a dependence on wire width. The spin-orbit scattering time increases as the width is reduced, as is also observed in wires fabricated from spin-orbit coupled two-dimensional systems in semiconductor heterostructures. The similarity is discussed in light of weak antilocalization in the two-dimensional strongly spin-orbit coupled Bi(001) surface states.