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MXenes stand out from other 2D materials because they combine very good electrical conductivity with hydrophilicity, allowing cost-effective processing as thin films. Therefore, there is a high fundamental interest in unraveling the electronic transport mechanisms at stake in multilayers of the most conducting MXene, Ti$_3$C$_2$T$_x$. Although weak localization (WL) has been proposed as the dominating low-temperature (LT) transport mechanism in Ti$_3$C$_2$T$_x$ thin films, there have been few attempts to model it quantitatively. In this paper, the role of important structural parameters -- thickness, interflake coupling, defects -- on the dimensionality of the LT transport mechanisms in spin-coated Ti$_3$C$_2$T$_x$ thin films is investigated through LT and magnetic field dependent resistivity measurements. A dimensional crossover from 2D to 3D WL is clearly evidenced when the film thickness exceeds the dephasing length $l_\phi$, which is in the $50-100\,\mathrm{nm}$ range. 2D WL can be restored by weakening the coupling between adjacent flakes, the intrinsic thickness of which is lower than $l_\phi$, hence acting as parallel 2D conductors. Alternatively, $l_\phi$ can be reduced down to the $10\,\mathrm{nm}$ range by defects. Our results clearly emphasize the ability of WL quantitative study to give deep insights in the physics of electron transport in MXene thin films.