We know that one is not permitted to make a bracha or say devarim shebikedusha in front of nakedness. Why then is it permitted for the mohel and father at a bris to make their berachos when the baby's erva is uncovered? Similarly, why may a bris be conducted in a shul, where there are sefarim (which are also not supposed to be in presence of erva)?
(if i am not wrong seforim is only a problem for relations not for being naked)
Shulchan Aruch Harav 75.7
The nakedness of an animal is not considered nakedness and it is permitted to recite the Shema in view of it. As to gentiles, even though of them it is written, “Their flesh is the flesh of donkeys,” their nakedness is considered nakedness, as [implied by] the verse, “They did not see the nakedness of their father,” and it is forbidden to recite [the Shema] in view of it. The same applies in view of the nakedness of a minor, even a one-day-old baby.
Some authorities permit [the Shema to be] recited in view of the nakedness of a minor as long as he is not old enough to engage in intimacy, i.e., until he is nine years old. The halachah follows this view. Therefore it is not necessary to cover the nakedness of a baby when reciting the blessing over the circumcision. There is, however, an authority who maintains that, nevertheless, the mohel not hold [the baby’s] nakedness while reciting the blessing.
the Shulchan Aruch Y"D 265.8 says you do not need to cover the erva when saying the brocho (but you should clean him is he dirtied his diaper (paraphrased))
the Tur brings the Baal Haitur that says it is not necessary to cover it since it is not Erva until he is fit for relations (which is 9 years old)
see B"Y there
the Kitur Yalkut Yosef 79.2 says the ervah of a child it is only a problem for shma not for blessings
also the Shulchan Aruch in O"H 75.4