This is a very interesting & complex subject. In my experience, these emergency turns are usually company specific, and mostly need to be followed at any time there is an engine failure. Day night VMC IMC irrelevant unless specified. The real problem is that ATC are not aware of the particular procedure for each airline and aircraft type.
When the crew is busy aviating and navigating, there might not be too much capacity to inform ATC of needed deviation from clearance. What if the frequency is in use just when you have the chance to make the call? Then what if there is conflicting traffic and you are told to maintain a heading or continue as cleared? Or "are you declaring an emergency?" "how many POB?" "what is your endurance?" "confirm you require to turn left onto heading xxx?" All this just as you're performing near max capacity to get everything else, including communication, done correctly.
I cannot believe we should need to inform ATC of our intentions in the event of an engine failure at every airfield where there is such a procedure. Try this at some foreign airfields and you'll have take-off clearance cancelled or delayed. Probably won't make that slot time. Too many times and you won't have a job.
We cannot ignore the fact that ATC should know in advance of the possibilities. The main point really raised by this post should be how we ensure they do know each single engine procedure for all companies and each of their aircraft types The difficulties don't stop there. If your particular procedure is day night VMC IMC specific, does everyone know which procedure? What if ATC think its night VMC and crew decide its still daylight, but IMC?
A situation I have faced is very out of date emergency turn procedure to a non existant navaid. Even if I do this, its harder to communicate my intention.
At another airfield, very soon after take-off an emergency turn was required over a close spaced parallel runway with exactly simultaneous military traffic on another frequency. The North Asian military controller had a very slippery grip of English, the military thought they were King and the local F/O doesn't want to make waves, no matter how well briefed. How is a controller going to make some traffic space when these guys have actually previously done a barrel roll around my A321 at 500ft just as I became visual on ILS final? If the aircraft is heavy, hot day and tailwind we will hit a hill by not turning. Just hope the military jet will break left to avoid a collision.
My life was greatly simplified by refusing to operate to that particular airfield and then retirement from aviation.
Those of you that have a long way yet to go, and who really care about such matters, have my sympathy. Try to solve this problem somehow.