When a particle changes its spin orientation is it instantaneous?

The spin evolution is continuous and deterministic (i.e. you have to evolve the state of the particle, or atom, containing information about its spin with the Schrodinger equation). In fact, Schrodinger equation is deterministic and the evolution will depend on the externally changing fields that couple to the spin (and possibly other degrees of freedom of the atom/particle).

It can be a messy evolution, especially in an atom where there are electron-electron interactions. For a single particle it's simpler (i.e. you have to consider the Schrodinger equation for a single electron in an external electro-magnetic field that can change in time).

However, as long as the external fields change continuously, the state of the particle (and its spin) will evolve continuously. Only if you "measure" the spin, then the state will collapse into a definite spin eigenstate (immediately, according to standard QM), but this has nothing to do with the changing magnetic field (it is simply the fact that the measurement procedure is not governed by the Schrodinger equation).

Answering the title:

When a particle changes its spin orientation is it instantaneous?

An elementary particle is a quantum mechanical entity which is described by the numbers in the table, including a fixed spin. The spin of a charged particle will be changing orientation when interacting with electric and magnetic fields. Even though the changes in direction obey classical electromagnetism, there is still the quantum mechanical energy-time uncertainty

that imposes a non instantaneous interaction, given the energies involved. The uncertainty principles are the envelopes of the possible exact solutions of a particular quantum mechanical interaction.

The same is true in all problems where there exists and intrinsic quantum mechanical total spin, as with composites like protons , or even atoms and molecules, i.e. where the quantum mechanical frame is necessary.