A bris can most certainly be held in a hospital, though very few are these days. You shall be circumcised in the flesh of your foreskin, and it shall be a sign of the covenant between Me and you. Then, as a member of the covenant community, the boy is given a Hebrew name, linking him to his Jewish family and to Jewish history. An authentic bris is done on the eighth day, as prescribed in the Torah. Doing the bris on the eighth day in the hospital involves going back into the hospital.
Many people believe that hospitals and doctors are more sanitary and safer.
Most of these are doctors or nurses who have been trained in the rituals of the bris. Finally, my sense is that if a child has already been circumcised but has not had a bris, most Reform rabbis would still do a ceremony bringing the child into the covenant and giving him a Hebrew name. We think it will be a girl. We want to do something to welcome her and give her a Hebrew name. Traditionally, a brit milah is the ceremony whereby a Jewish boy is brought into the covenant.