The compound CuTe (vulcanite) undergoes a quasi one dimensional charge density wave (CDW) at $T< T_{\mathrm{CDW}}=335$ K with a $5\times1\times2$ periodicity. The mechanism at its origin is debated. Several theoretical works claimed that semilocal functionals are unable to describe its occurrence and ascribed its formation only to strong electron-electron interaction. Moreover, the possible role of quantum anharmonicity has not been addressed. Here, by performing quantum anharmonic calculations, we show that semilocal functionals correctly describe the occurrence of a CDW in CuTe if ultradense electron momentum grids allowing for small electronic temperatures are used. The distortion is driven by the perfect nesting among 1D Fermi surface sheets extending in the $k_y$ direction. Quantum anharmonic effects are important and tend to suppress both the distortion and $T_{\mathrm{CDW}}$. The quantum anharmonic structural minimization of the CDW phase in the generalized gradient approximation leads, however, to distorted Te-Te bond lengths in the low temperature phase that are $21\%$ of the experimental ones at $T=20$ K. This suggests that, even if the electron-electron interaction is not crucial for the mechanism of CDW formation, it is relevant to accurately describe the structural data for the low-T phase. We assess the effect of correlation on the CDW by using the DFT+U+V approximation with parameters calculated from first principles. We find that correlation enhances the Te-Te distortion, T$_{CDW}$ and the total energy gain by the distortion.