At the beginning, I’m referring only to the classical point of view and not to the quantum field theory (developed for the inner-atom interactions), for which I hope to see the answer from other people.
We often hear that the electromagnetic wave "consists" of real photons while the electromagnetic field "consists" of virtual photons.
What is an EM field? A field is something that exert a force. An EM field does not exert any force. What we have are electric fields and magnetic fields. Electric fields exert force between charged particles and magnetic fields exert force on the magnetic dipole moments of the subatomic particles. About the third case, the Lorentz force and the over induction processes see the following description of Lorentz force below.
What is a EM wave? Each photon is particle with an oscillating electric and an oscillating magnetic field component. If the sources - mostly electrons - get accelerated synchronous the emitted photons are in phase too and the resulting radio wave is really a wave with oscillating field components. The emission of photons from a thermic source one hardly can call a wave, nether you’ll be able to measure directly a wave characteristic from a thermic source.
Since the idea of field lines is the only model for electric and magnetic fields and the inner structure of these fields (field lines) is not developed, the only possibility how one can explain the interaction in these fields are virtual photons. Although, during the approach of an electron to the nucleus there are realized real photons and real photons are involved to excite electrons from the nucleus.
When we see a trajectory of an electron deflected in the magnetic field, we can (with the above stated understanding) hypothetically describe this interaction as the electron absorbing and/or emitting virtual photons (interacting with the field created by other charges).
Let me go into details. A moving - non-parallel to an external magnetic field - electron gets aligned with its magnetic dipole moment to the external field, by this gets deflected, that is an acceleration, emits a real photon (see Synchrotron radiation, by this gets disaligned, and so on until the electron exhausts its kinetic energy and comes to rest in the center of the spiral of its trajectory. This phenomenon is called the Lorentz force.
We also often hear that the electromagnetic wave involves alternating electric and magnetic fields. If an electron crosses the wave (which could be a low frequency), it seems that the electron would also be deflected.
If an electron is under the influence of EM radiation some of the photons of this radiation would interact with the electron. Simply the electron absorbs the photon and the electron gain energy. Or nothing happens because photons are indivisible particles from their emission until their absorption. (You will not be able to change the wavelength of any EM radiation without absorption and re-emitting processes.) So in general I agree with your statement below:
It seems that the only most probable process is the Compton scattering that is dramatically different from an electron simply being deflected in the magnetic or electric field with no real photons to scatter.
The next of your statements will have the answers above, if you are willing to rename “EM field” and “EM wave” by “EM radiation”. I’ve changed the expressions as follows:
Is his really the case that the EM radiation is different in nature from the static field produced by local charges? For example, is the moving electron deflected differently in the static magnetic field compared to a low enough frequency radio wave?
Yes. That is how th receiver of an antenna works.
Is it true that the deflection in the static magnetic field produces no scattered photons...
No. See the synchrotron radiation, the photons are not scattered, but simple emitted from the exhausting electron.
... while the deflection in the otherwise similar field of a low frequency radio wave results in a bunch of scattered photons?
The absorption of the photons from EM radiation is accompanied be the re-emission of photons of different wavelengths. The radio wave gets damped.