Once airbourne you still have these crossed controls. The aircraft will still be heading in the direction of the runway centreline, while sideslipping to the right (into wind). This sideslip will equal the crosswind and the two cancel out. The aircraft is therefore tracking the runway centreline.
Now at this point all you need to do now is slowly relieve the pressure on the rudder pedals until they have naturally come back to the centre. When you say:
Blip, once airborne, almost immediately you have to reduce rudder input to zero. I have found that the aeroplane will not weathercock enough into wind to counter the drift by itself. You see this when watching planes sliding downwind after take-off when they haven't applied enough drift- very few ever move upwind. So my answer to Snakes initial query in the first post:
Would you use rudder during aircraft rotation in crosswind conditions to push the nose into the wind? This in purpose to assure a correct drift angle to maintain runway track after lift-off.
Or...., wait until after lift-off and then bank slightly into the wind to maintain runway track.
....was
not to then use rudder into wind to apply further drift required but to bank slightly towards the crosswind keeping balanced flight until you have the full heading offset required to maintain runway track.