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The merger of two or more galaxies can enhance the inflow of material from galactic scales into the close environments of Active Galactic Nuclei (AGN), obscuring and feeding the supermassive black hole (SMBH). Both recent simulations and observations of AGN in mergers have confirmed that mergers are related to strong nuclear obscuration. However, it is still unclear how AGN obscuration evolves in the last phases of the merger process. We study a sample of 60 Luminous and Ultra-luminous IR galaxies (U/LIRGs) from the GOALS sample observed by NuSTAR. We find that the fraction of AGN that are Compton-thick (CT; $N_{\rm H}\geq 10^{24}\rm\,cm^{-2}$) peaks at $74_{-19}^{+14}\%$ at a late merger stage, prior to coalescence, when the nuclei have projected separations of $d_{\rm sep}\sim 0.4-6$ kpc. A similar peak is also observed in the median $N_{\rm H}$ [$(1.6\pm0.5)\times10^{24}\rm\,cm^{-2}$]. The vast majority ($85^{+7}_{-9}\%$) of the AGN in the final merger stages ($d_{\rm sep}\lesssim 10$ kpc) are heavily obscured ($N_{\rm H}\geq 10^{23}\rm\,cm^{-2}$), and the median $N_{\rm H}$ of the accreting SMBHs in our sample is systematically higher than that of local hard X-ray selected AGN, regardless of the merger stage. This implies that these objects have very obscured nuclear environments, with the $N_{\rm H}\geq 10^{23}\rm\,cm^{-2}$ gas almost completely covering the AGN in late mergers. CT AGN tend to have systematically higher absorption-corrected X-ray luminosities than less obscured sources. This could either be due to an evolutionary effect, with more obscured sources accreting more rapidly because they have more gas available in their surroundings, or to a selection bias. The latter scenario would imply that we are still missing a large fraction of heavily obscured, lower luminosity ($L_{2-10}\lesssim 10^{43}\rm\,erg\,s^{-1}$) AGN in U/LIRGs.