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Type Ia supernovae (SNe Ia) were instrumental in establishing the acceleration of the universe's expansion. By virtue of their combination of distance reach, precision, and prevalence, they continue to provide key cosmological constraints, complementing other cosmological probes. Individual SN surveys cover only over about a factor of two in redshift, so compilations of multiple SN datasets are strongly beneficial. We assemble an up-to-date "Union" compilation of 2087 cosmologically useful SNe Ia from 24 datasets ("Union3"). We take care to put all SNe on the same distance scale and update the light-curve fitting with SALT3 to use the full rest-frame optical. Over the next few years, the number of cosmologically useful SNe Ia will increase by more than a factor of ten, and keeping systematic uncertainties subdominant will be more challenging than ever. We discuss the importance of treating outliers, selection effects, light-curve shape and color populations and standardization relations, unexplained dispersion, and heterogeneous observations simultaneously. We present an updated Bayesian framework, called UNITY1.5 (Unified Nonlinear Inference for Type-Ia cosmologY), that incorporates significant improvements in our ability to model selection effects, standardization, and systematic uncertainties compared to earlier analyses. As an analysis byproduct, we also recover the posterior of the SN-only peculiar-velocity field, although we do not interpret it in this work. We compute updated cosmological constraints with Union3 and UNITY1.5, finding weak 1.7--2.6 sigma tension with LambdaCDM and possible evidence for thawing dark energy (w0 > -1, wa < 0). We release our SN distances, light-curve fits, and UNITY1.5 framework to the community.