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Yes some delicate situation. Here we have to take a lot of information into consideration. Actually electrons move at random direction and in the absence of an electric field the net velocity of the electrons will be zero. Hence we can assume the initial velocity to be zero. Now as an electric field is applied then a force of eE is acting on the electron opposite to the field direction. Due to this electrons get accelerated. No doubt about it. But this acceleration is in effect only for a duartion known as relaxation time. Hence the speed picked up will be acceleration x relaxation time.

So this velocity attained in that duration is called as Drift velocity.

Vd = (eE / m )* t.

Here m is mass of electron.

Hence a constant velocity Vd

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Q: When there exists an electric field in a conductor why do electrons move with constant velocity but not with an acceleration?
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