We know that electrons flow in a circuit due to the electric field, so shouldn't the electrons be accelerating, but we also know that $i=neAv$ so from here we can conclude $v=i/neA$ which is a constant, and also if the electrons are accelerating in the circuit, according to Maxwell they should also generate EM waves, thus lose energy, right? Does the first part of the question have something related to Drude's equation?
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Electrons are scattered by the lattice imperfections, phonons, other electrons, etc. One way to see it is as electron being accelerated by the electric field for some time, and then being scattered in an arbitrary direction, and losing the extra velocity it gained. As simpler (although less correct description) is by seeing electrons as balls moving in a viscous liquid - they are accelerated by the electric field, but slowed down by friction and end up having a constant velocity (called drift velocity.)
Electrons do emit some energy, although most energy is lost in their collisions and transferred to the lattices. In some special cases the part of energy lost via emission can be significant, as, e.g., in X-ray tube. In this relation note that this is not the case in incandescent lamp, where the emission is mostly due to the filament being heated by the electric current flowing through it (see, detailed discussion in How is light emitted by an incandescent lamp?)
You may find more mathematical description (in terms of simplified Drude model) in this answer: Heating effect of electric current.
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$\begingroup$ So, is it that they have some acceleration at start then they collide so they retard and due to field so they again accelerate, this keeps going on, so it's like we average the velocity to be equal to drift velocity. Right? $\endgroup$ Mar 4 at 9:16
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