Different operators may have different SOP's but I imagine that the main factor is the certification type of the A/C. In types that do not guarantee a rate of climb after an engine failure (all singles and many small twins) you keep the hand on the levers as you will be looking to securing the failed engine ASAP. In types that guarantee minimum climb gradients after an engine failure (FAR/JAR25 ?) you move both hands to the control wheel at V1. Here the priority is controling the A/C and climbing away safely. In my company we do nothing until we reach 400' and then the emphasis is on securing the correct engine ie.- don't rush it.
SP and MC I do not feel is as important. If I was flying a Cessna Caravan in a multicrew environment I would keep one hand on the thrust lever, after all you would not even have a v1.