I wake my LO and go.
If we need to be somewhere, we plan to leave about 20 minutes earlier than actually needed just to allow time for figuring things out.
If DD is sleeping, I get myself completely ready (including diaper bag, water, etc) then I wake her with enough time to change her diaper and clothes and nurse if needed. I try to at least "top her off" with milk to make things easier as well. Thankfully my LO is soothed by the movement of the car, so she usually falls back to sleep as soon as we get going.