When a brit milah (circumcision) is performed on Yom Kippur, there are adjustments made to the ceremony.
First, the mohel performs the mesisa on Yom Kippur. During this part of the ceremony, the mohel draws out blood from the wound itself after the milah. Although technically drawing blood is a biblical prohibition, since mesisa is done for the health of the child, it is done on Shabbat and Yom Kippur. Some mohalim have the custom of putting a bit of wine in their mouth before the mesisa; that should not be done on Yom Kippur.
Second, it is customary to say the borei peri hagefen blessing during the milah, before the naming of the baby. The Shulhan Arukh (Yoreh De’ah 265:4) writes that the blessing over the wine is not said on Yom Kippur. However, the custom of many communities, such as Saloniki, Damascus, Halab, and Egypt, was to make the blessing and give the wine to a minor, who is not obligated to fast, to taste. Some even had the custom to make the beracha and give a drop of the wine to the baby; R. Ovadia Yosef objected to this practice, as the wine should be given to a ben da’at. In practice (see Hazon Ovadia, Yamim Noraim pg. 340), the borei peri hagefen blessing is said, and the wine is given to a minor.
Finally, our custom is to say the blessing on besamim on Yom Kippur. There is an additional advantage, as it helps us fulfill the misvah of reciting me’ah berachot (one hundred blessings) each day. Therefore, it is permissible to say the beracha over a hadas branch on Yom Kippur.
Summary: When a brit milah is performed on Yom Kippur, the mohel does the mesisa but does not put any wine into his mouth. The blessing over the cup of wine is said, but the wine is given to a minor, who is not obligated to fast, to taste. The blessing over the besamim is said on Yom Kippur.