Magnetic field dependent universal conductance fluctuations (UCF's) are observed in weakly disordered indium tin oxide nanowires from 0.26 K up to $\sim 25$ K. The fluctuation magnitudes increase with decreasing temperature, reaching a fraction of $e^2/h$ at $T \lesssim 1$ K. The shape of the UCF patterns is found to be very sensitive to thermal cycling of the sample to room temperatures, which induces irreversible impurity reconfigurations. On the other hand, the UCF magnitudes are insensitive to thermal cycling. Our measured temperature dependence of the root-mean-square UCF magnitudes are compared with the existing theory [C. W. J. Beenakker and H. van Houten, Phys. Rev. B \textbf{37}, 6544 (1988)]. A notable discrepancy is found, which seems to imply that the experimental UCF's are not cut off by the thermal diffusion length $L_T$, as would be expected by the theoretical prediction when $L_T < L_\varphi$, where $L_\varphi$ is the electron dephasing length. The approximate electron dephasing length is inferred from the UCF magnitudes and compared with that extracted from the weak-localization magnetoresistance studies. A reasonable semiquantitative agreement is observed.Comment: 11 pages, 9 figure