We present new near-infrared (NIR) Cepheid period-Wesenheit (P-W) relations in the LMC using time-series observations from the Large Magellanic Cloud NIR Synoptic Survey. We also derive optical+NIR P-W relations using V and I magnitudes from the Optical Gravitational Lensing Experiment. We employ our new JHK_s_ data to determine an independent distance to the LMC of {mu}_LMC_=18.47+/-0.07 (statistical) mag, using an absolute calibration of the Galactic relations based on several distance determination methods and accounting for the intrinsic scatter of each technique. We also derive new NIR period-luminosity and Wesenheit relations for Cepheids in M31 using observations from the Panchromatic Hubble Andromeda Treasury survey. We use the absolute calibrations of the Galactic and LMC W_J,H_ relations to determine the distance modulus of M31, {mu}_M31_=24.46+/-0.20 mag. We apply a simultaneous fit to Cepheids in several Local Group galaxies covering a range of metallicities (7.7<12+log[O/H]<8.6 dex) to determine a global slope of -3.244+/-0.016 mag/dex for the W_J,H_ relation and obtain robust distance estimates. Our distances are in good agreement with recent TRGB based distance estimates and we do not find any evidence for a metallicity dependence in the NIR P-W relations.