Did Mary stay with Elizabeth till the birth of John the Baptist? We read at Luke 1:36 that Elizabeth was six months pregnant when she was visited by Mary. We also read at Luke 1:56 that Mary stayed with Elizabeth for nearly three months. Read together, these two verses mean that Elizabeth had either completed or was in the final month of her pregnancy the day Mary went back to her home.  One is, therefore, curious to know if Mary stayed with Elizabeth till John was born. Evangelist Luke, having once been a doctor, ought to have highlighted that Mary stayed with Elizabeth till the birth of the latter's  son, just to emphasize the helping hand Mary had extended to her . My question is: Are there any Catholic traditions which conclusively state that Mary in deed stayed with Elizabeth till the birth of John the Baptist ?
Cornelius à Lapide, S.J.'s commentary on Luke 1:56:

There is a question here whether the Blessed Virgin remained until the nativity of John. Theophylact, Euthymius, Jansenius maintain that she did not.
They prove this

because the Blessed Virgin came in the sixth month, and remained, as it were, three months (Vulg. quasi tres menses), therefore she departed before the ninth month was completed, and therefore before Elizabeth gave birth to the child;
because after the departure of the Virgin, S. Luke relates the birth of the child, without making any mention of the Virgin;
and chiefly, because it was not fitting that the Virgin should be present at the birth;
because it was proper that she should avoid the crowd which would gather together at the birth.

The contrary opinion is equally, and even more, probable; and is maintained by Origen, S. Ambrose, &amp;c.—

because it would have been discourteous to remain up to the birth and leave immediately before it;
because at the time of the birth Elizabeth most needed the presence, help, and consolation of the Virgin;
because the Virgin remained three months for this very purpose, that she might behold, embrace, and bless John, the wonder of the world, and the forerunner of Christ, and on the other hand that John when born might behold and venerate the mother of the Lord, and Christ the Lord in her. Whence Bede says, “Mary remained until, Elizabeth’s full time having come, she saw the nativity of the forerunner of her Lord, on account of whom chiefly she had come.”


Private revelation among the mystics holds this latter opinion, too. See the end of ch. 9 of The Life of Mary as Seen by the Mystics.
Direct quotes from St. Ambrose, Theophylact, and St. Bede appear in St. Thomas Aquinas's Catena Aurea on Luke 1:56:

St. Ambrose: Mary abode with Elisabeth until she had accomplished the time of her bringing forth; as it is said, And Mary abode, &amp;c.
Theophylactus: For in the sixth month of the conception of the forerunner, the Angel came to Mary, and she abode with Elisabeth three months, and so the nine months are completed.
St. Ambrose: Now it was not only for the sake of friendship that she abode so long, but for the increase also of so great a prophet. For if at her first coming the child had so far advanced, that at the salutation of Mary he leaped in the womb, and his mother was filled with the Holy Spirit, how much must we suppose the presence of the Virgin Mary to have added during the experience of so long a time? Rightly then is she represented as having shown kindness to Elisabeth, and preserved the mystical number.
St. Bede: For the chaste soul which conceives a desire of the spiritual word must of necessity submit to the yoke of heavenly discipline, and sojourning for the days as it were of three months in the same place, cease not to persevere until it is illuminated by the light of faith, hope, and charity.
Greek exegesis: For it is the custom for virgins to go away when the pregnant woman brings forth. But when she reached her own home, she went to no other place, but abode there until she knew the time of her delivery was at hand. And Joseph doubting, is instructed by an Angel.