
Not urgent. A root adds diameter every year and concrete does not move aside, so it comes down to which is stronger, and a live structural root beats four inches of unreinforced slab. This is a property-damage and trip-hazard problem, not a tree-failure problem, and the tree is usually in good health. The important inversion is that the repair is what creates the hazard: the roots lifting your driveway are the buttress roots on one side of the trunk, and cutting them is precisely how a stable tree becomes an unstable one.
Watch. The lift advances at the tree's growth rate — a fraction of an inch a year on a mature tree — so nothing about the tree becomes dangerous while you decide. What does change is the liability: the trip lip gets taller, and where the walk is public sidewalk, maintenance responsibility often falls to the adjoining property owner (this varies by municipality and is UNVERIFIED for your address — confirm with your city). Separately, water in the crack plus freeze-thaw will break the slab further each winter faster than the root will.