"If I make the logic inside of the elevator actor, how do I begin the timeline sequence? "
inside the elevator actor, you can make a custom event called ElevatorMoveToNextFloor. inside the widget you can make a variable of type ElevatorActor reference, and set it to editable and exposed on spawn. then in the elevatorActor, you can spawn the widget, and pass itself as a reference.
then when you click on a button in the widget, you can call the ElevatorMoveToNextFloor function on the ElevatorActor Reference that was set on spawn.
its hard to explain with words. you just need more practice with communicating between blueprints. watch this: