Measure the distance between the tip of the existing stair nose and the riser.
Cut a piece of plywood that is the same thickness as the measurement between the riser and the stair nose to the same size of the riser; install it on the riser to make the stair nose flush with the riser.
Measure the width of the bottom riser and cut your first plank of hardwood flooring to this width.
Apply a thin, serpentine line of non-water-based construction adhesive to the first plank and press it into place on the bottom of the riser -- make sure to leave any needed gaps for expansion.
Continue cutting planks for the first riser and installing in this manner. Any planks you've ripped should be installed with the cut at the top of the riser so it will be hidden by the stair nose.
Top nail each plank on the riser around the edges and toward the middle.
Cut a piece of flush stair nose to the width of your stair. Make sure you account for necessary expansion gaps.
Install the stair nose by apply a thin, serpentine bead of construction adhesive and pressing it into place. Top nail the piece along the sides and in the middle.
Measure and cut the first plank for the first tread to the width of the stair, again taking into account the necessary expansion gaps.
Apply a thin, serpentine bead of construction adhesive and fit the first plank into the stair nose. Hold the stair nose and first plank together with painter's tape.
Measure and cut other planks for the treads, installing with the construction adhesive and top nailing. Install any planks you've ripped with the cut side at the back of the tread facing the riser.
Continue this process as you work your way up the staircase.