A metal conducts electricity because of the pool of free electron which move from higher potential to lower poetntial.
So, when we connect a battery to a metal wire, the potential difference is created which leads the electrons to move from higher potential to lower until the charge neutralizes and we say that the battery is dead.
Velocity = I/nAq
where V = drift velocity,
n = number of charge carriers
A = Area of cross section
q = charge
I = current.