When you say carpet runner, do you mean the width of the carpet will inlay into finished wooden stair threads. ?? Or will the stairs be completely carpeted. Picture? I'm not exactly picking up what your puttin down.
There is an OSB product which is used for stair threads. It's about 11" - 12" deep and 1" thick, has a rounded nosing and can can be cut to various lengths.
A small nosing could be attached using glue and screws, however this depends on the construction and materials used in the existing stairs. Best to have a solid thread which creates a nosing IMO.