It actually wouldn't. The bottom couple of kilometers would come down, the rest would burn up on reentry or go shooting out into deep space, depending on where it gets cut. Everything above the cut goes zipping off into deep space, everything below falls back into the atmosphere. Everything above ~50km is going to end up going fast enough to burn up on in the atmosphere.
You'd need to put the tether at the equator, so you'd probably attach it to a large floating platform. Attach some thrusters and you've got some rudimentary maneuverability to avoid collisions in the first place.
A severed orbital elevator would be very expensive, but it's not going to cause any serious damage back on Earth.