Does heat radiation come from the nucleus or the electrons? Is more of the thermal radiation due to acceleration of electrons or acceleration of the nuclei? Do electrons and nuclei carry the same fractions of thermal energy in a hot body? 
 A: In neutral atoms and molecules, heat cannot usually come from their constituents except at very high temperatures. These constituents, electrons and nucleus, are mutually bound quantum mechanically at a specific energy level and cannot radiate because they usually are at the lowest possible energy level.
Radiation from the constituents of the atoms that will contribute to heat directly may come from scatterings on each other, and if the energy is enough electrons on the outer shells may be kicked to a higher level by the impact and then decay to a lower level by electromagnetic radiation. In this sense it is the electrons that take part in the radiation, although in a center of mass system one could say the same for the nucleus-and-rest-of-the-electrons, because the energy levels are the solutions of specific potential problems.
At room temperatures most of the radiation coming from heat is due to the spill over fields of the atoms and molecules that give rise to van der Waals forces . Again the kinetic energy of the atoms  in collisions , raises individual atoms to higher collective energy  in the spill over fields, and the return to ground state releases electromagnetic ratiation.
At the micro level it is quantum mechanical frameworks that are necessary for the details.

Do electrons and nuclei carry the same fractions of thermal energy in a hot body?

The atoms and molecules as one entity carry the kinetic energy corresponding to the thermal energy. Their constituents cannot be connected with heat.
A: 
Do electrons and nuclei carry the same fractions of thermal energy in a hot body?

It's nearly all in the electrons. For example, in hydrogen, the momentum of the proton and electron are equal and opposite (in the c.m. frame), so $K=p^2/2m$ is thousands of times larger for the electron. You might be tempted to imagine that there's $(3/2)kT$ worth of energy in the proton and $(3/2)kT$ in the electron, but that would violate conservation of momentum. The fundamental principle is that you get $(1/2)kT$ per degree of freedom, but because of conservation of momentum, the hydrogen atom only has 3 d.f., not 6.

Is more of the thermal radiation due to acceleration of electrons or acceleration of the nuclei?

It's almost all from the electrons. The Larmor formula for the radiated power (averaged over a cycle) is $P=(2/3)kq^2a^2/c^3$. The accelerations of the electrons are thousands of times larger.
