Are Electrons in a Circuit Subject to Newton's Third Law?

Yes, conservation of angular momentum applies and the system will not rotate. I assume that the bulb radiates isotropically and also that any other radiation effects are isotropic. The electrons cannot be set into motion without a reaction force on the battery. However the magnetic field will polarise electron spins, free or located in ferromagnetic atoms. This will align their spin angular momentum possibly causing an undoubtedly tiny opposite angular momentum of the setup.

More importantly any orbital angular momentum of the electrons will be transferred (back) to the lattice. There should be a transient effect on the angular orientation when the current is changing.

Newton's laws do apply. The overall system will not rotate. While the electrons are rotating in one direction, the rest of the gear will indeed be rotating imperceptibly slowly in the other. But the rotation is constant, as the propulsive EMF is being exactly opposed by collision forces between electrons and metal atoms.

I would suggest that these collisions would also destroy any alignment of electron spins, negating any cumulative effect there (But I am not that much the physicist, so I can't really argue that case). A shame, as an experiment, to see if (mass) electron spin alignment is reflected in a counteracting macroscopic material angular momentum, would be an interesting one.