We present photometric and morphological analyses of nuclear star clusters (NSCs)-very dense, massive star clusters present in the central regions of most galaxies-in a sample of 33 massive disk galaxies within 20 Mpc, part of the "Composite Bulges Survey." We use data from the Hubble Space Telescope including optical (F475W and F814W) and near-IR (F160W) images from the Wide Field Camera 3. We fit the images in 2D to take into account the full complexity of the inner regions of these galaxies (including the contributions of nuclear disks and bars), isolating the NSC and bulge components. We derive NSC radii and magnitudes in all three bands, which we then use to estimate NSC masses. Our sample significantly expands the sample of massive late-type galaxies with measured NSC properties. We clearly identify NSCs in nearly 80% of our galaxies, putting a lower limit on the nucleation fraction in these galaxies that is higher than previous estimates. We find that the NSCs in our massive disk galaxies are consistent with previous NSC mass-NSC radius and galaxy mass-NSC mass relations. However, we also find a large spread in NSC masses, with a handful of galaxies hosting very low-mass, compact clusters. Our NSCs are aligned in PA with their host galaxy disks but are less flattened. They show no correlations with bar or bulge properties. Finally, we find the ratio of NSC to BH mass in our massive disk galaxy sample spans a factor of ~300.