Why doesn't the nucleus have "nucleus-probability cloud"?

Yes, the nucleus is composed of subatomic particles that have a probability cloud. Protons and neutrons fill orbitals in the nucleus just like electrons in the atom do. What's more, every proton or neutron is a complex particle itself and the quarks inside have their very own probability cloud. (Quarks are simple objects that have no internal structure as far as we know.)

Uncertainty principle requires that the nucleus as a whole has some spatial spread.

The easy part is that the "probabilistic cloud" of a nucleus and its constituents are way smaller than the space electrons pretend to occupy. That's what makes the point approximation viable.

Very often indeed the nucleus is assumed motionless. It is then assumed that the motion of the nuclei and the electrons can be treated separately. This is known as the Born-Oppenheimer approximation. The reason is that solving the equations simultaneously is very difficult and would not be very efficient.

Note that for the hydrogen atom this approximation is not required. In this two particle case the wave function describes the relative motion and position of electron and nucleus.

The nucleus does have a probability cloud. As the simplest example, consider the hydrogen-1 atom. Conservation of momentum requires that the center of mass of the electron and proton remain fixed. Therefore we have

$$\Psi_p(\textbf{x}|=(\text{const.})\Psi_e(-\alpha\textbf{ x}),$$

where $\Psi_p$ is the wavefunction of the proton, $\Psi_e$ is the wavefunction of the electron, and $\alpha$ is the ratio of the masses. Because $\alpha$ is large, one can often approximate the proton as being fixed at one point.