The interferometer mirrors of gravitational-wave detectors (GWDs) are Bragg reflectors made of alternate amorphous silica (SiO$_2$) and titania-doped tantala (TiO$_2$:Ta$_2$O$_5$) layers as low- and high-refractive index materials, respectively. A thermal treatment is usually performed to reduce both mechanical losses and near-infrared (NIR) optical absorptions of the coatings. The authors present a spectroscopic ellipsometry (SE) investigation of the effect of annealing and Ti:Ta mixing on Ta$_2$O$_5$ coatings deposited under conditions similar to those adopted for building up mirrors of GWDs. The broadband analysis covers both the NIR and the fundamental absorption threshold region. The data show an evident annealing-induced reduction of the fundamental optical absorption broadening. Modeling the data through the Cody-Lorentz formula confirms that NIR absorptions are below the SE sensitivity and shows a notable annealing-induced reduction of so-called Urbach tails. Titania-doping of tantala slightly reduces the Urbach energy. After the heating treatment, the resulting Urbach energy of the doped material is lower than that of annealed pure tantala. The observed reduction of Urbach tails is important because it parallels the reduction of the so-called internal friction observed in mechanical measurements so that SE emerges as a convenient tool for an agile diagnostic of both the optical and the mechanical quality of amorphous oxide coatings.