How London dispersion forces come into existence out of nowhere?

The explanation of the force as due to oscillating dipoles is only an approximate description and should be regard only as a guide for students. The actual explanation is that the wavefunctions of the two atoms overlap and the system now needs to be described by a larger wavefunction that includes the electrons in both atoms.

In principle this new wavefunction could be calculated directly by numerical methods, but as computers were not available to London in 1930 he did the calculation using perturbation theory. In any case, the energy of the new wavefunction is slightly lower than the energies of the two atoms when separated to infinity, so we find there is a small attractive force between the atoms. For more on this see the Wikipedia article on the subject.

I will stick my neck out and suggest that we can justify the oscillating dipole explanation as follows:

When the two atoms are separated to infinity the electrons in them are in eigenstates of the Hamiltonian and these eigenstates are time independent i.e. there are no oscillations in their charge densities.

However when we bring the atoms together the interaction between them changes the eigenstates so the electrons in the atoms are no longer in eigenstates but would be in a superposition of the eigenstates. In general a superposition of energy eigenstates is not time independent and such superpositions will have oscillations in the charge density. These oscillations will generate the fluctuating electric dipoles used in the heuristic description of the force.

So if you look at the situation this way it is the interaction between the two atoms, i.e. the electrostatic forces between the electrons and nucleus in atom A with the electrons and nucleus in atom B, that causes the oscillating dipoles to appear.

Nothing. The atomic cloud ( electron distribution) is probabilistic in nature. Untertainty in charge distribution causes polarization