FGM, at least the dominant form practiced in Egypt, West, Saharan, and the Horn of Africa is pre-Christian and a relict tradition most likely originating as an early proto-Afroasiatic socio-religious rite.
As concerning Christian cultures and their practice for FGM, pre-Islamic Christian Egyptians most certainly - as evidenced by widespread current practice - widely practiced FGM, and the same for the pre-Islamic Beja/Cushitic and Nubian Christian societies. Coptic Egyptians still observe the practice today, and FGM was observed in both Graco-Roman and Byzantine Egypt by Greek and Byzantine commentators as being widespread [1].
Almost all Ethiopian Christian societies also universally practiced FGM [2], equally amongst the Ethiosemitic Amhara, Tigray, and Tigringya Christians and Cushitic Agew, Oromo, and Beja, and still do today. Also, the Beta Israel jewry are the only Jewish group to practice FGM [3].
The tradition has no association with either Islam or Christianity, and is a substrate tradition of pagan rites which in many societies - both Christian and Muslim - has given no exclusive religious association and is more seen through a cultural and social lens than a religious one.
[1] Knight, Mary. "Curing Cut or Ritual Mutilation?: Some Remarks on the Practice of Female and Male Circumcision in Graeco-Roman Egypt." Isis 92.2 (2001): 317-38. Web.
[2] Belmaker, R. "PL02 Female Genital Mutilation: Successful Social Change Exemplified by Israeli Bedouin and Ethiopian Jews." Asian Journal of Psychiatry 4 (2011)
[3] N. Grisaru, S. Lezer, R.H. Belmaker "Ritual female genital surgery among Ethiopian Jews." Archives of Sexual Behaviour, Volume 26, 1997, p. 2