When an atom has an electron in an excited energy level and it transitions to a lower level it emits a photon. What direction is it likely to emit the photon in? Are all directions equally likely, even toward the nucleus?
In the general case, "no" as there is a angular momentum transfer involved (meaning there are preferred directions relative the original and/or final angular momentum of the atom).
That said, for most matter at room temperature the atoms have random orientation so you can treat the answer as "yes" for experimental purposes.
Now, I see that you are wondering about the nucleus in the question. There are two thing to keep in mind here.
First you may be thinking that the photon is emitted by the electron and that the electron is on a particular side of the atom at the time of emission. (That is, you may have some version of the almost completely incorrect Rutherford/Bohr/de Broglie atom in your head.)
It is the system of nucleus and electron(s) that undergoes a change and emits a photon, and even from the beginning asking "from what part of the atom is the photon released?" has no precise answer.
Second the nucleus is also a quantum system and can't absorb arbitrary amounts of energy in internal changes (it wouldn't be able to absorb it in arbitrary translational motion either because that would violate the conservation of momentum---it drags the electron with it, after all). The nuclear energy splittings are generally too large to absorb an atomic photon.
It depends on the mechanism by which the photon is emitted.
- Stimulated Emission: Yes, the emitted photon will inherit the characteristics of the photon that stimulated it, including its propagation direction. That's how lasers get coherent light.
- Spontaneous Emission: No, this should be random orientation. Another way to think about spontaneous emission is as stimulated emission where the stimulating photons are virtual photons, who have random orientation themselves.