We present new near-infrared Cepheid Period-Wesenheit relations in the LMC using time-series observations from the Large Magellanic Cloud Near-Infrared Synoptic Survey. We also derive optical$+$near-infrared P-W relations using $V$ and $I$~magnitudes from OGLE-III. We employ our new $JHK_s$ data to determine an independent distance to the LMC of $\mu_{\rm LMC} = 18.47\pm0.07 {\textit{(statistical)}}$~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 near-infrared Period-Luminosity and Wesenheit relations for Cepheids in M31 using observations from the PHAT survey. We use the absolute calibrations of the Galactic and LMC $W_{J,H}$ relations to determine the distance modulus of M31, $\mu_{\rm M31} = 24.46\pm0.20$~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 -$3.244\pm0.016$~mag/dex for the $W_{J,K_s}$ 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 near-infrared P-W relations.