The requirements in quantum optics experiments for high single-photon detection efficiency, low timing jitter, low dark count rate and short dead time have been fulfilled with the development of superconducting nanowire single-photon detectors. Although they offer a detection efficiency above 90%, achieving a high time resolution in devices made of amorphous materials is a challenge, particularly at temperatures above 0.8 K. Devices made from niobium nitride and niobium titanium nitride allow us to reach the best timing jitter but, in turn, have stronger requirements in terms of film quality to achieve a high efficiency. Here we take advantage of the flexibility of reactive co-sputter deposition to tailor the composition of Nb_xTi_1-xN superconducting films and show that a Nb fraction of x = 0.62 allows for the fabrication of detectors from films as thick as 9 nm and covering an active area of 20 µm, with a wide detection saturation plateau at telecom wavelengths and in particular at 1550 nm. This is a signature of an internal detection efficiency saturation, achieved while maintaining the high time resolution associated with NbTiN and operation at 2.5K. With our optimized recipe, we reliably fabricated detectors with high critical current densities reaching a saturation plateau at 1550 nm with 80% system detection efficiency and with a FWHM timing jitter as low as 19.5 ps.