Consider a long, straight wire of cross-sectional area A carrying a current i. Let there be n free electrons per unit volume. An observer places himself on a trolley moving in the direction opposite to the current with a speed 
and separated from the wire by a distance r. The magnetic field seen by the observer is very nearly.
We know, drift velocity of electron is
vd = i/(nAe)
Where
i is the current in the wire,
n is the electron number density of the wire a,
A is the area of cross section of the wire,
e is the charge of the electron.
We can see that the trolley is moving with drift velocity but current travels in a conductor with a speed comparable to that of light in free space. So the velocity of the trolley is very low compared to the velocity at which the current travels. So there will be no significant change in the value of B and it will be 
.