Does the nucleus also revolve around the centre of mass of atom? If yes then will it radiate any energy?why or why not
Why the down-votes? The nucleus does revolve around the center of mass. Do electrons radiate? In classical physics both should, but neither do.
The resolution to the paradox comes in two steps. 1) use center-of-mass (COM) coordinates and relative coordinates 2) use quantum mechanics. The system as a whole is neutral so the motion of the COM does not radiate (roughly speaking).
That leaves the relative motion of the electrons and nucleus. The relative coordinates takes into account the positions of both the electrons and the nucleus. The electronic transitions that occur in an atom are transitions of the system as a whole, not the electrons alone. So the motion of the nucleus is accounted for in the quantum mechanical picture.
Note that the motion of the nucleus is also taken into account in the classical picture if the same change of coordinates is used. The problem with that is that the classical picture predicts radiation that is not observed in real life. If the classical picture were correct, the nucleus would indeed radiate.
Some words about the revolving. The now outdated idea about a rotation of electrons around the nucleus comes from the experimental setup: Atoms were bombed with electrons and as a result the atom seems to have a tiny nucleus and a lot of empty space around with orbiting electrons. The orbiting was dismissed but the name gets a transformation into orbits and means the probability of finding an electron in a specific region around the atom's nucleus.
The nucleus interacts with electromagnetic radiation. For example you can break or accelerate or hold in one place a nucleus with beams of light. And any interaction with EM radiation is not only an absorption of photons but also a radiation of photons. For this radiation it is not necessary to have the nucleus under rotation.