Exclusion principle in electron orbitals If a 1s or 2s orbital is to be occupied by two electrons, their spins must be antiparallel. If we start with single occupancy: if a second electron is added to the orbital (as, for instance, in a chemical reaction), how does the first electron convey to the second what its spin orientation is? And what is the mechanism by which the two electrons then adjust their spin orientations to achieve the antiparallel spin state?
 A: 
If a 1s or 2s orbital is to be occupied by two electrons, their spins must be antiparallel. If we start with single occupancy: ... how does the first electron convey to the second what its spin orientation is?

If an explanation like from Wikipedia


Despite sometimes being called an exchange force in analogy to classical force, it is not a true force, as it lacks a force carrier.


is not really helpful there is an alternative image for you.
Every electron has a magnetic dipole moment and the spin is only another expression for this phenomenon of the electron.$^{*)}$ Now imagine how two bar magnets stick together in the “lowest energy state”:

That is an easy imagination of why only two electrons - with opposite orientation of their magnetic dipole moments- can occupy the same state in an atom.

And what is the mechanism by which the two electrons then adjust their spin orientations to achieve the antiparallel spin state?

The influence of the magnetic behavior of electrons inside atoms is neglected at the moment. You could think about the orientation of the tiny magnets or you can hold in mind the Pauli exclusion principle without additional explanation.

*) Magnetic moment of an electron on Wikipedia:

... an electron indeed behaves like a tiny bar magnet. One consequence is that an external magnetic field exerts a torque on the electron magnetic moment depending on its orientation with respect to the field.

