When a current carrying conductor is placed inside magnetic field, what motion does it take? Does the motion not occur and only Hall emf is obtained. Or if there is a motion due to Lorentz Force, magnetic force can't do work. So it must be electric field doing the work. How does electric field do work to move the conductor. P.S . Supposing that the conductor is always connected externally to Voltage for a current flow.
The classic experiment discussing the magnetic force is 2 wires carrying currents in the same direction will pull each other and 2 wires carrying opposite currents repel each other - meaning from the beginning of scientific research in electromagnetism, it was known that magnetic fields do apply a force on currents (before the discovery of the Hall effect). You are correct that the magnetic force is actually applied on the moving electrons and not the bulk itself, so the question is indeed valid.
First of all, from Newtonian mechanics we can understand that when the net force on the wire isn't $0$ (because its electrons feel the Lorentz force) the center of mass of the system will accelerate so if the electrons can't leave the bulk, you can see how the bulk needs to accelerate as well. But that exactly is the point - the electrons can't leave the bulk, they are electrically attached to it, meaning the electric field the electrons apply on the positive stationary charges is the one responsible for moving the entire wire (and not only its electrons) - this is the origin of the kinetic energy the bulk gains.