In my experience of giving multi engine training using mixture cuts the engine starts immediately when the mixture is returned to rich...unless you have feathered the prop etc...which you generally don't do.
While the mixture was used to give the initial 'failure' as soon as the student had correctly identified the failure and carried out the memory items I would reinstate the mixture and set zero thrust on the now running engine.
The MOST important point of all is where you fail the engine not how. Any really experienced multi piston pilot uses 'gear up' as their decision point...i.e. if the engine fails before gear up I close both throttles and land straight ahead irrespective..after gear up it is memory items and fly if possible (and most DO despite the rhetoric). Of course it is a little situation specific and a decision will (should) have been made before commencing the takeoff based on knowledge of the aeroplane and the ambient conditions. A light twin at MTOW on a 40deg day will not fly on one so you would be predisposed to a controlled forced landing before even setting takeoff power.
That many multi instructors feel the need to fail engines right on lift off with the gear still down indicates ignorance and inexperience.
On the odd occasion when I have needed to climb into a piston twin with an unknown individual in the RHS I have made it ABUNDANTLY clear that if they fail an engine on me below 200' with the gear down I WILL land straight ahead with NO compunction whatsoever...If I bend metal it's their problem. Whether it was my tone of voice or the look in my eyes I am not sure but none have ever tried it on.
Morons have caused piston twins to have the reputation they have not the aircraft themselves. ANY piston twin flown correctly is 'safer' than a single. But that is the rub...since most piston twins passed from the hands of 3rd level airlines to small charter companies 15-20 years ago there has been a HUGE loss of corporate knowledge on how they should be flown.
The average CASA FOI is part of the problem not the solution.