If we consider a simple dc circuit with just a resistor, we notice that V is constant and so is the length through which current is flowing, hence the electric field is constant. So why does drift velocity also remain constant shouldn't it be increasing linearly?
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Why do you think drift velocity should be increasingly linearly ? In a resistive material the drift velocity $u$ is proportional to the electric field $E$, so if $E$ is constant so is $u$ (ignoring any transient state when the current is first turned on). See the Wikipedia article on drift velocity for further details.
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