After a merger of two massive black holes (MBHs), the remnant receives a gravitational wave (GW) recoil kick that can have a strong effect on its future evolution. The magnitude of the kick ($v_\mathrm{recoil}$) depends on the mass ratio and the alignment of the spins and orbital angular momenta, therefore on the previous evolution of the MBHs. We investigate the cosmic effect of GW recoil by running for the first time a high-resolution cosmological simulation including GW recoil that depends on the MBH spins (evolved through accretion and mergers), masses and dynamics computed self-consistently. We also run a twin simulation without GW recoil. The simulations are run down to $z=4.4$. We find that GW recoil reduces the growth of merger remnants, and can have a significant effect on the MBH-galaxy correlations and the merger rate. We find large recoil kicks across all galaxy masses in the simulation, up to a few $10^{11}\,\rm M_\odot$. The effect of recoil can be significant even if the MBHs are embedded in a rotationally supported gaseous structure. We investigate the dynamics of recoiling MBHs and find that MBHs remain in the centre of the host galaxy for low $v_\mathrm{recoil}/v_\mathrm{esc}$ and escape rapidly for high $v_\mathrm{recoil}/v_\mathrm{esc}$. Only if $v_\mathrm{recoil}$ is comparable to $v_\mathrm{esc}$ the MBHs escape the central region of the galaxy but might remain as wandering MBHs until the end of the simulation. Recoiling MBHs are a significant fraction of the wandering MBH population. Although the dynamics of recoiling MBHs may be complex, some retain their initial radial orbits but are difficult to discern from other wandering MBHs on radial orbits. Others scatter with the halo substructure or circularise in the asymmetric potential. Our work highlights the importance of including GW recoil in cosmological simulation models.