  We present time-series observations of Population \rom{2} Cepheids in the
Large Magellanic Cloud at near-infrared ($JHK_s$) 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 $K_s$-band data in the Galactic Bulge
from VVV 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 {\it
Gaia DR1} for VY~Pyx and the {\it Hubble Space Telescope} parallaxes for
$k$~Pav and 5 RR Lyrae variables to obtain an absolute calibration of the
Galactic $K_s$-band P-L relation, resulting in a distance modulus to the LMC of
$\mu_{\rm LMC} = 18.54\pm0.08$~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 the distances to these globular
clusters based on Pop~II Cepheids are consistent (within $2\sigma$) with
estimates based on the $M_V-[\rm{Fe}/\rm{H}]$ relation for horizontal branch
stars.
