The Rama (Rav Moshe Isserles of Cracow, 1530-1572) writes that it is customary for a Mohel to serve as the Hazan in the synagogue on the day he performs a Berit. The Machshireh Mila (by Hacham Eliyahu Shama Ha’levi, Chief Rabbi of Aleppo, d. 1814) brings an explanation for this practice from an earlier source (listen to audio recording for precise citation), noting that serving as Hazan in the synagogue is akin to a Kohen serving in the Bet Ha’mikdash. The Talmud Yerushalmi relates that in ancient times, when somebody would be invited to lead the prayer service in the synagogue, he would be told, "Come bring our sacrifice!" – clearly indicating that a Hazan leading the prayers is compared to the Kohen in the Mikdash offering the sacrifices. Now circumcision, in a sense, is a type of "sacrifice," and thus the Mohel, who performs the circumcision, is like the Kohen serving in the Mikdash. It is therefore appropriate for him to also serve as the Hazan, leading the prayer service, which is similarly akin to ministering in the Bet Ha’mikdash.
The Machshireh Mila adds that the word "Mila" itself alludes to this practice, as it may be read as acrostic for the words, "Mohel Yered Lifneh Ha’teba" ("The Mohel shall go before the ark"), referring to the role of Hazan.
It must be noted, however, that, as the Shach (Rav Shabtai Ha’kohen, 1621-1662) comments, if a mourner is also present in the synagogue, he is granted precedence over the Mohel. As we know, it is customary for a mourner to lead the prayer service as a source of merit for the deceased, and this practice overrides the custom that a Mohel leads the prayer service on the day he performs a Berit.
Summary: It is customary for a Mohel to lead the prayer service in the synagogue on the day he performs a Berit, however, if a mourner is also present in the synagogue, the mourner is granted this privilege over the Mohel.