  Computing thermal transport from first-principles in UO$_2$ is complicated
due to the challenges associated with Mott physics. Here we use irreducible
derivative approaches to compute the cubic and quartic phonon interactions in
UO$_2$ from first-principles, and we perform enhanced thermal transport
computations by evaluating the phonon Green's function via self-consistent
diagrammatic perturbation theory. Our predicted phonon lifetimes at $T=600$ K
agree well with our inelastic neutron scattering measurements across the entire
Brillouin zone, and our thermal conductivity predictions agree well with
previous measurements. Both the changes due to thermal expansion and
self-consistent contributions are nontrivial at high temperatures, though the
effects tend to cancel, and interband transitions yield a substantial
contribution.
