Mussaf is always done during CHM in phreigish/Yom Tov nusach; on HHM Shabbos Mussaf, the additions for shabbos are done in Shabbos amidah nusach.
The yom tov musaf prayers post-Amidah are said, as Sam said. The high holiday kaddish titkabal of Yankel der Hezeriker is used after hoshanos (many of us use this after all of mussaf shalosh regalim); I’ve even heard some people use yamim noraim nusach for pezukei dezimrah, barechu, and michamocha.