The spin relaxation time of photoinduced conduction electrons has been measured in InGaN quantum disks in GaN nanowires as a function of temperature and In composition in the disks. The relaxation times are of the order of 100 ps at 300 K and are weakly dependent on temperature. Theoretical considerations show that the Elliott-Yafet scattering mechanism is essentially absent in these materials and the results are interpreted in terms of the D'yakonov-Perel' relaxation mechanism in the presence of Rashba spin-orbit coupling of the wurtzite structure. The calculated spin relaxation times are in good agreement with the measured values.