The Rite of Profession of Religious (who are not simultaneously receiving the Consecration of Virgins) is contained in the Roman Ritual which is open to all priests. Religious do not have to be virgins and widows were never permitted to receive the veil from the hands of the bishop. Yes, a priest can say the prayer of blessing for all religious because religious life is not the same thing as consecrated virginity. The consecration of religious occurs at the profession of vows made before a superior which means that it is the superior (lay or clerical) who mediates the consecration by virtue of receiving the vows.
The Rite of Consecration of Virgins is contained in the Roman Pontifical, which is the liturgical book for bishops. Only bishops can validly impart the consecration of virgins to virgins in the world and this has always been the case since apostolic times. Only female virgins may receive this consecration. If you ever read in this House of Brede, the pratagonist does not receive the consecration of virgins with the others in her class because she had a child out of wedlock, but I think she eventually became abbess. The Rite has two versions; one is for religious and the other is for women living outside religious institutes. Starting from the Blessed Virgin Mary, there are no centuries to my knowledge which lacked consecrated virgins. The Order of Virgins is alive and strong. It is the fastest growing vocation for women in the universe. Since 1970 when the Rite opened up for women living in the world, there have been an estimated 6,000-7,000 consecrations which is very impressive considering that the largest female religious order (Salesians) have an estimated 17,000 members total.
Religious life requires the vows of evangelical counsels, communal life, and separation from the world. Chastity is vowed from that time forward whereas sacred virginity is resolved to be maintained perpetually but no vow is made to that effect by women living in the world. The consecration of virgins is prayed by the bishop and is not the effect of vows being made by the women. Virgins living in the world do not make vows. Their consecration occurs like confirmation or ordination- it is a spiritual anointing mediated by the bishop by the prayer of consecration over them.