Look online for videos of a slinky toy being dropped. The top starts falling when it is released, but the bottom does not move yetthe bottom does not move yet! The bottom is still supported by material that is not itself supported. The loss of tension is noticed by other parts of the system at the speed of sound in that material.
The space elevator, assuming it’s under tension and cut at the bottom, is the same thing. Various cars gripping it will notice the lack of tension after a delay ed on the speed of sound. Likewise, this is the time scale at which it will start to bend and whip around.
A ground observer can’t see the tension. He won’t notice anything until it moves far enough to resolve. How fast the cut end snaps up depends on the specific design and tension, and can vary by orders of magnitude! Newer concepts don’t need an anchor at alldon’t need an anchor at all, but hang in the air and normally don’t touch the ground. So at one extreme you won’t see anything happen; at the other it will shoot up at supersonic (in air) speed.