First sand or cut the round nose of each stair square. We will put a new round nose on as the last step.
Clean the stair off to bare wood, or at least to very solid varnish.
Cut stair coverings out of new prefinished laminate flooring. These are fairly thin and will not affect the height of the steps very much. Cut battens to temporarily take the place of the nose of each step, and cover them with wax paper.
Apply PL Premium construction adhesive to the entire stair and use it to fill in the depression in the worn old stair. Why this adhesive? It dries rock hard, has no solvents to smell bad, does not shrink at all, and will never break up under the force of walking up and down the stairs. Seat the laminate flooring into the glue, assuring that there is total support, no voids under the laminate.
Quickly scrape off any excess adhesive from the front of the step and set the wax paper covered batten against the riser and the edge of the new laminate, screwing it into the riser. The screw hole will be covered later by the round nose trim. This will stop the PL Premium from running out the front of the step. Work from the top stair down, so that any glue drips will not end up on the new laminate.
A day later, remove the batten and glue on a hardwood round nose that goes up high enough to cover the front of the laminate. This can be sanded down for a perfect fit and then stained and finished. Use the same PL Premium to glue this on, but try to avoid a large glue line here because it will show up dark brown. You may want to tack the nose into place with finishing nails to keep the joint tight while the glue dries.
You may want to choose a contrasting colour for the nosing to help people with a visual impairment distinguish the front of each step.