aaahhhh !! mea maxima culpa .. I misunderstood the question to relate to AEO (brain infarct when one considers the rest was clearly OEI) .. if we are talking OEI for this as well .. of course, one has a rudder input until time permits to trim it out. Other things being equal aim to climb out with around 2-3 deg bank into the operating engine(s) and rudder as required ... ask legions of turboprop pilots .. the DTs have nothing on the shaking OEI leg takeoff ..
... maybe I need a good night's sleep ...
Re deviation .. nil wind is the general rule .. if you want guarantees, let me offer you one .. a min V1 departure with a maximum crosswind .. will just about guarantee you a trip in the weeds before you know where you are .. the case is not considered at certification and the pilot is expected to exercise a bit of discretion in whether he should go in these circumstances ..
.. retires muttering to himself in rambling terms ....