The ATLAS Forward Proton (AFP) detector is intended to measure protons scattered at small angles from the ATLAS interaction point. To this end, a combination of 3D Silicon pixel tracking modules and Quartz-Cherenkov time-of-flight (ToF) detectors is installed 210m away from the interaction point at both sides of ATLAS. Beam tests with an AFP prototype detector combining tracking and timing sub-detectors and a common readout have been performed at the CERN-SPS test-beam facility in November 2014 and September 2015 to complete the system integration and to study the detector performance. The successful tracking-timing integration was demonstrated. Good tracker hit efficiencies above 99.9% at a sensor tilt of 14{deg}, as foreseen for AFP, were observed. Spatial resolutions in the short pixel direction with 50 {mu}m pitch of 5.5 +/- 0.5 {mu}m per pixel plane and of 2.8 +/- 0.5 {mu}m for the full four-plane tracker at 14{deg} were found, largely surpassing the AFP requirement of 10 {mu}m. The timing detector showed also good hit efficiencies above 99%, and a full-system time resolution of 35 +/- 6 ps was found for the ToF prototype detector with two Quartz bars in-line (half the final AFP size) without dedicated optimisation, fulfilling the requirements for initial low-luminosity AFP runs.