Yes but “smoke in the cockpit” takes 1 second to transmit.
I’ve personally transmitted this myself while turning back to the airfield without ATC permission.
Fotunately, they granted permission to land any runway without my declaring an emergency.
Not saying that you are wrong here.
It is not about time, it is priority. We don’t talk if we are busy with extremely important procedures. A fire in the flight deck will often have zero indication to ATC until well into the emergency procedure. I know, I train this scenario.
Would the pilots has to look up the new way point,or would it be easier to contact ATC for a vector?