Black holes (BHs) are ubiquitous components of the center of most galaxies. In addition to their mass, the BH spin, through its amplitude and orientation, is a key factor in the galaxy formation process, as it controls the radiative efficiency of the accretion disk and relativistic jets. Using the recent cosmological high-resolution zoom-in simulations, N EW H ORIZON and G ALACTICA , in which the evolution of the BH spin is followed on the fly, we have tracked the cosmic history of a hundred BHs with a mass greater than 2 × 10 4 M ⊙ . For each of them, we have studied the variations of the three-dimensional angle (Ψ) subtended between the BH spins and the angular momentum vectors of their host galaxies (estimated from the stellar component). The analysis of the individual evolution of the most massive BHs suggests that they are generally passing by three different regimes. First, for a short period after their birth, low-mass BHs ( M BH < 3 × 10 4 M ⊙ ) are rapidly spun up by gas accretion and their spin tends to be aligned with their host galaxy spin. Then follows a second phase in which the accretion of gas onto low-mss BHs ( M BH ≲ 10 5 M ⊙ ) is quite chaotic and inefficient, reflecting the complex and disturbed morphologies of forming proto-galaxies at high redshifts. The variations of Ψ are rather erratic during this phase and are mainly driven by the rapid changes of the direction of the galaxy angular momentum. Then, in a third and long phase, BHs are generally well settled in the center of galaxies around which the gas accretion becomes much more coherent ( M BH > 10 5 M ⊙ ). In this case, the BH spins tend to be well aligned with the angular momentum of their host galaxy and this configuration is generally stable even though BH merger episodes can temporally induce misalignment. We even find a few cases of BH-galaxy spin anti-alignment that lasts for a long time in which the gas component is counter-rotating with respect to the stellar component. We have also derived the distributions of cos(Ψ) at different redshifts and found that BHs and galaxy spins are generally aligned. Our analysis suggests that the fraction of BH-galaxy pairs with low Ψ values reaches maximum at z ∼ 4 − 3, and then decreases until z ∼ 1.5 due to the high BH-merger rate. Afterward, it remains almost constant probably due to the fact that BH mergers becomes rare, except for a slight increase at late times. Finally, based on a Monte Carlo method, we also predict statistics for the 2D projected spin-orbit angles λ . In particular, the distribution of λ traces the alignment tendency well in the three-dimensional analysis. Such predictions provide an interesting background for future observational analyses.