  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.
