In 2013 April a new magnetar, SGR 1745-2900, was discovered as it entered an outburst, at only 2.4 arcsec angular distance from the supermassive black hole at the Centre of the Milky Way, Sagittarius A*. SGR 1745-2900 has a surface dipolar magnetic field of ~ 2x10^{14} G, and it is the neutron star closest to a black hole ever observed. The new source was detected both in the radio and X-ray bands, with a peak X-ray luminosity L_X ~ 5x10^{35} erg s^{-1}. Here we report on the long-term Chandra (25 observations) and XMM-Newton (8 observations) X-ray monitoring campaign of SGR 1745-2900, from the onset of the outburst in April 2013 until September 2014. This unprecedented dataset allows us to refine the timing properties of the source, as well as to study the outburst spectral evolution as a function of time and rotational phase. Our timing analysis confirms the increase in the spin period derivative by a factor of ~2 around June 2013, and reveals that a further increase occurred between 2013 Oct 30 and 2014 Feb 21. We find that the period derivative changed from 6.6x10^{-12} s s^{-1} to 3.3x10^{-11} s s^{-1} in 1.5 yr. On the other hand, this magnetar shows a slow flux decay compared to other magnetars and a rather inefficient surface cooling. In particular, starquake-induced crustal cooling models alone have difficulty in explaining the high luminosity of the source for the first ~200 days of its outburst, and additional heating of the star surface from currents flowing in a twisted magnetic bundle is probably playing an important role in the outburst evolution. ; Comment: 16 pages, 12 figures; accepted for publication on MNRAS