Well this clearly involves some airmanship and opinions may vary somewhat, and this is simply one of many:
On DEP: the emergency turns in rwy analysis are usually for obstacle avoidance and they must be briefed with every departure and followed in IMC. Turns mandated by noise abatement and other 'cosmatic' factors are void and null with emergency declaration. The general practice is to first fly the airplane, do recall items, clean up, reduce thrust to MCT, trim, then call ATC let alone accepting turns. If contacted by ATC in the meantime, decalre emergency and say you'll call back.
Normally they don't give vectors before positive control and 1000' AGL. But if there is emergency turn, you have to include it in your procedure, all the more reason to spend the shortest time on the radio. Once all established, normally ATC helps out by giving vectors for a landing or to your TO alternate as the case may be. If possible, it is probably better to take longer legs on vectors, DME hold, turns on the good engine,... as to opposed to normal tight hold.
On GA however, it is different and one HAS to follow the procedure to avoid obstacles or worse yet collision, unless of course you have time to ask for rwy heading 3000', which is always the best for ENG out TO and GAs.
As for the original question, A ONE ENG INOP procedure ends when the airplane is safely on the ground and PAX/CREW are either evacuated or deplaned!