Luminous, compact stellar systems known as nuclear clusters (NCs) are commonly found in the centers of galaxies across the entire Hubble sequence. I present an analysis of the structure and stellar populations of a sample of ten of the nearest and brightest NCs residing in late-type spiral galaxies, using images from Hubble Space Telescope Wide Field Camera 3 in seven bands that span the near-ultraviolet to the near-infrared in wavelength. The intrinsic shapes and sizes of the NCs, disentangled from the effects of point spread function (PSF) blurring, were measured using GALFIT. We find evidence for radial color gradients within the NCs, as well as young disk structures aligned with the host galaxy disk. In color-color diagrams spanning the near-UV through the near-IR, NCs tend to lie far from single-burst evolutionary tracks, indicating the presence of multi-age populations.I developed a Monte Carlo code to fit linear combinations of simple stellar population models to the observed spectral energy distribution (SED) of each NC and assess the uncertainties in the fit parameters. Tests on a catalog of mock SEDs demonstrate that our method gives unbiased mass age, and reddening estimates for populations with U -- V colors redder than ∼ --2 mag. Stellar masses computed via SED fitting are in good agreement with previous dynamical studies. The NCs are generally dominated by an old (> 1 Gyr) population component, but are best described by temporally extended star formation histories. On average, populations with ages