We present time-series observations of Population II Cepheids in the Large Magellanic Cloud at near-infrared (JHKs) wavelengths. Our sample consists of 81 variables with accurate periods and optical (VI) magnitudes from the OGLE survey, covering various subtypes of pulsators (BL Herculis, W Virginis, and RV Tauri). We generate light-curve templates using high-quality I-band data in the LMC from OGLE and Ks-band data in the Galactic bulge from VISTA Variables in Via Láctea survey and use them to obtain robust mean magnitudes. We derive period-luminosity (P-L) relations in the near-infrared and Period-Wesenheit (P-W) relations by combining optical and near-infrared data. Our P-L and P-W relations are consistent with published work when excluding long-period RV Tauris. We find that Pop II Cepheids and RR Lyraes follow the same P-L relations in the LMC. Therefore, we use trigonometric parallax from the Gaia DR1 for VY Pyx and the Hubble Space Telescope parallaxes for k Pav and 5 RR Lyrae variables to obtain an absolute calibration of the Galactic Ks-band P-L relation, resulting in a distance modulus to the LMC of mag. We update the mean magnitudes of Pop II Cepheids in Galactic globular clusters using our light-curve templates and obtain distance estimates to those systems, anchored to a precise late-type eclipsing binary distance to the LMC. We find that the distances to these globular clusters based on Pop II Cepheids are consistent (within ) with estimates based on the relation for horizontal branch stars.