In presence of lectern field do all electrons have same average velocity?

vd = a τ

where α is the acceleration due to the electric field

 τ is the average relaxation time. 

Now 

Acceleration of each electron is,  α = -eE/m

=> vd  = -eEτ/m

Electric field in a conductor doesn't change for a given value of potential V. 

At constant temperature, average relaxation time remains unchanged.

Therefore, the drift velocity remains constant for constant temperature.