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We obtained Hubble Space Telescope/Wide Field Camera 3 imaging of a sample of ten of the nearest and brightest nuclear clusters (NCs) residing in late-type spiral galaxies, in seven bands that span the near-UV to the near-IR. Structural properties of the clusters were measured by fitting two-dimensional surface brightness profiles to the images using GALFIT. The clusters exhibit a wide range of structural properties, with F814W absolute magnitudes that range from -11.2 to -15.1 mag and F814W effective radii that range from 1.4 to 8.3 pc. For 6 of the 10 clusters in our sample, we find changes in the effective radius with wavelength, suggesting radially varying stellar populations. In four of the objects, the effective radius increases with wavelength, indicating the presence of a younger population that is more concentrated than the bulk of the stars in the cluster. However, we find a general decrease in effective radius with wavelength in two of the objects in our sample, which may indicate extended, circumnuclear star formation. We also find a general trend of increasing roundness of the clusters at longer wavelengths, as well as a correlation between the axis ratios of the NCs and their host galaxies. These observations indicate that blue disks aligned with the host galaxy plane are a common feature of NCs in late-type galaxies, but are difficult to detect in galaxies that are close to face-on. In color-color diagrams spanning the near-UV through the near-IR, most of the clusters lie far from single-burst evolutionary tracks, showing evidence for multi-age populations. Most of the clusters have integrated colors consistent with a mix of an old population (>1 Gyr) and a young population (∼100-300 Myr). The wide wavelength coverage of our data provides a sensitivity to populations with a mix of ages that would not be possible to achieve with imaging in optical bands only. The surface brightness profiles presented in this work will be used for future stellar population modeling and dynamical studies of the clusters.