For the teetering head, there's no reason for the fuselage to tilt with the rotor head directly.
This is the way I think of it:
Cyclic input makes disc tilt, and tilted lift vector drags the disc off in the desired direction.
Point of attachment of mast to disc is the rotor head, so as the disc flies off into the sunset, the head comes with it.
Pendulously dangling fuselage, attached to mast, starts to move off in the same direction. However, as speed builds up, drag means the fuselage hangs back with respect to the head, hence the body angle change we see from the cockpit.
In short, push the cyclic forward, and shortly thereafter the deck angle will change too as you fly off in that direction.
With a more rigid head, the control power leverage as described by Crab above will make things happen quicker.