We analyze Hubble Space Telescope observations of six globular clusters in the Large Magellanic Cloud from program GO-14164 in Cycle 23. These are the deepest available observations of the LMC globular cluster population; their uniformity facilitates a precise comparison with globular clusters in the Milky Way. Measuring the magnitude of the main sequence turnoff point relative to template Galactic globular clusters allows the relative ages of the clusters to be determined with a mean precision of 8.4%, and down to 6% for individual objects. We find that the mean age of our LMC cluster ensemble is identical to the mean age of the oldest metal-poor clusters in the Milky Way halo to 0.2 ± 0.4 Gyr. This provides the most sensitive test to date of the synchronicity of the earliest epoch of globular cluster formation in two independent galaxies. Horizontal branch magnitudes and subdwarf fitting to the main sequence allow us to determine distance estimates for each cluster, and examine their geometric distribution in the LMC. Using two different methods, we find an average distance to the LMC of 18.52 ± 0.05.