Space elevator is not something which stands on the ground and grows up. It is something which lives on geostationary orbit and grows down from it to the Earth. If you cut the lower 1 meter of it, the rest will hang...
...that is, until a satellite will hit the cable above. Space elevator is built in the equatorial plane, all satellites cross it, so eventually every satellite is going to collide with the cable. For this reason the space elevator is incompatible with existing spaceflight, that's why even with nanotubes it's unlikely to be built.