Symmetry adapted expressions for the magnetic and quadrupolar nuclear spin-lattice relaxation time T1 in HCP metals are derived from Dirac theory. They are applied systematically to 3d, 4d and 5d metals (Sc, Ti, Y, Zr, Tc, Lu, Hf, Re, Os) on the basis of semi-relativistic and full-relativistic self-consistent LMTO calculations. General trends and relativistic effects are discussed. The relaxation rate T-11 in Zr is overestimated theoretically by a factor of two, as already found by Asada and Terakura. There is strong evidence that the reason for this error lies in spin-orbit splitting. Shifts of 3 mRy in the band position explain the experimental T-11 in Zr. The Fermi surface of Ti is discussed in connection with the theoretical relaxation rate. Some quantities that are useful for the evaluation of measurements like hyperfine coupling constants and ratios between magnetic and quadrupole relaxation rates are presented. Comparison with the experimental data shows that the quadrupole scattering is well reproduced by the theory.