Will applying drywall directly over nail plates create a noticeable bulge? I am not applying any texture to the wall. If so, what is the best way to avoid any bulge? How far should drywall screws be from the nail plates to prevent cracking the drywall?
Will applying drywall directly over nail plates create a noticeable bulge?
It depends on the circumstances. If the wall has a lot of oblique natural light and/or will have a gloss finish, yes the bulge will likely show. Depending on how critical the need for a perfect wall is, should determine your efforts to eliminate the bulge.
what is the best way to avoid any bulge?
There are several tricks, with preference likely being opinion based. I have found that the easiest/quickest way is to mark plate locations with a pencil and use my electric planer to shave the stud back at plate locations so the plates sit flush or less than flush. Since your plates are already nailed in, you will need to pop them off to do it, but it is much easier that trying to hog out the back of the plasterboard.
It looks like you've got way more than 1-1/4" space between the face of the stud and the wires, so your plates may not be necessary by code.
Having said that, 6" up and down for screws will work, and you probably won't see a bulge. (Unless you run around with a straightedge, drywall is surprisingly forgiving. Out of plane studs are usually more of a problem than plates.) You can whack the plates a little with a hammer to force them a little more flush.