When we provide a potential difference in a circuit it does nothing more than provide an electric field to the conductor,When an electric field is provided in a conductor the free electrons move to the opposite direction of the field and hence there is an accumulation of charge on the ends of the conductor.Due to the accumulation of charges,the negative charge and the positive charges produce an internal electric field on the opposite direction of the external electric field.At some point these will be of equal magnitude and the Enet will be zero.So,how will electrons move in such as situation and have drift velocity?
Your description is correct, and this is exactly why open circuits don't work.
If you connect a conductor to one end of a battery but keep the other end free, then charges will accumulate at the far end setting up an opposing electric field. Soon, no charges will move anymore.
In order to keep a steady current that doesn't stop at some point, you cannot allow accumulation of charge anywhere. You must take the charges away from that other end as they arrive. You take them away by attaching this other end to the other battery pole.
A battery is an electrochemical "machine" which moves charges from one pole to the other (roughly speaking). By letting charges arrive at the opposite battery pole, they will be moved back to the starting point, ready to take another lap through the circuit.
There is now no accumulation but a circular loop of charge flowing (with their drift velocity) in steady current, as race cars on a racing track. Without a closed track, you will indeed see accumulation somewhere in the circuit, and charge flow will eventually (this happens extremely quickly) stop.
What you are describing sounds like a battery connected to a wire where the other end of the wire is not connected to the other battery terminal. Or it could also describe the charging of a capacitor by a voltage source (e.g., battery). The battery voltage removes electrons from one plate of the capacitor and deposits them on the other plate unit the accumulation of charge on the plates creates a voltage equal and opposite to the battery voltage. Then the current stops.
In general, however, charges don't accumulate in conductors in a complete electrical circuit unless there is an electrical energy storage device. The voltage source, such as a battery, continues to supply electrical potential energy to the charges so that current continues to flow.
Hope this helps.
The situation you described is seen in case of conductors when net E inside it becomes 0 due to external field and induced field
In such cases as the field is 0 the electric potential on the surface becomes constant and the surface is called equipotential surface due to which there isnt any motion of charges which require a potential to move