I’ve used a hole saw to do similar then you can fix a batten to the back face of the plasterboard through the hole and use the piece you cut out to fill the hole with a small amount of filler around the edges and in the screw heads.
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Easier than fitting a batten behind to secure the chopped out panel, use your jab saw to cut the edge at 45 degree angle, so when replacing the panel it rests against this angled edge and doesn’t fall back into the cavity.
The other approach I’ve seen is a cut-out slightly lower exposing the top and bottom of the noggin, then chiseling a trench in to lay the cable in (rather than a hole through the middle of the noggin), before replacing the panel against the noggin.
As above. Note the hole only needs to be big enough to get the drill bit in not the whole drill. Maybe better for you to cut out a small recess, as Arfa suggests, to pass the cable through the nogging. Remember to keep cable runs in line and vertical or horizontal to the sockets and switches to easily identify the cable runs later, diagonal cable runs are not allowed.
Something like that. What we need are some pics of the work as it progresses so we can so we can tut and say how we wouldn’t have done it that way. 
What can possibly go wrong? Time to get stuck in!
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Just like cutting the top of a pumpkin at haloween
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