If I do a Taharah have I also fulfilled the Mitzvoh of Halvayat Hamet, even if after the Taharah I don’t go to the gravesite or even wait for the actual Levayah to begin?

The mitzvah of levaya is to accompany the deceased 4 amot on his way to the burial place. Tahara is a great mitzvah but not the mitzvah of levaya. If the leveya did not begin and you need to leave you don’t need to wait.