The magnetotransport in a set of identical parallel AlGaN/GaN quantum wire structures was investigated. The width of the wires was ranging between 1110 nm and 340 nm. For all sets of wires clear Shubnikov--de Haas oscillations are observed. We find that the electron concentration and mobility is approximately the same for all wires, confirming that the electron gas in the AlGaN/GaN heterostructure is not deteriorated by the fabrication procedure of the wire structures. For the wider quantum wires the weak antilocalization effect is clearly observed, indicating the presence of spin-orbit coupling. For narrow quantum wires with an effective electrical width below 250 nm the weak antilocalization effect is suppressed. By comparing the experimental data to a theoretical model for quasi one-dimensional structures we come to the conclusion that the spin-orbit scattering length is enhanced in narrow wires.