Did Mary have the assistance of a midwife at the time of Jesus' birth? What do the traditions of Catholic Church say about it?
The short answer is eemaha! That is right eemaha, which is Eskimo for maybe!
The Gospels are silent on this point and the Church has not pronounced on this subject matter.
The Apocryphal Gospel of St. James states that a midwife came to the assistance of Mary’s birth of Jesus, but she arrived too late and the Infant Jesus was already born. Although the Church does not recognize this book in her biblical canon, it is the very source she uses for the names of the parents of the Virgin Mary: Joachim and Anne!
(1) And I saw a woman coming down from the mountain and she said to me, "Man,
where are you going?"
(2) And I said, "I am seeking a Hebrew midwife."
(3) Replying, she said to me, "Are you from Israel?"
(4) And I said to her, "Yes."
(5) Then, she said, "And who is giving birth in the cave?"
(6) And I said, "The one who has pledged to be married to me."
(7) And she said to me, "She is not your wife?"
(8) And I said to her, "She is Mary, the one who was raised in the temple. I won her by lot to be my wife. (9) She is not yet my wife, but has a fetus from the Holy Spirit."
(10) And the midwife said, "Really?"
(11) And Joseph said to her, "Come and see."
(12) So the midwife went with him. (13) And they stood near the cave and a dark cloud was hovering over the cave. (14) And the midwife said, "My soul glorifies this day, for today my eyes have seen a miracle: salvation has come to Israel."
(15) And immediately, the cloud withdrew from the cave and a great light appeared in the cave so that their eyes could not bear it. (16) And a little while later the same light withdrew until an infant appeared. And he came and took the breast of his mother, Mary.
(17) And the midwife cried out and said, "How great this day is for me, for I have seen this new miracle."
(18) And the midwife departed from the cave and met Salome and said to her, "Salome, Salome, I have to describe this new miracle for you. A virgin has given birth, although her body does not allow it."
(19) And Salome said, "As the Lord my God lives, unless I insert my finger and investigate her, I will not believe that a virgin has given birth."
Blessed Catherine Emmerich revelations suggest that no midwife attended the actual birth of Jesus. Thus Mother and Child were alone and totally at peace with one another.
Birth of the Child Jesus
I saw Joseph on the following day arranging a seat and couch for Mary in the so-called Suckling Cave of Abraham, which was also the sepulcher of Maraha, his nurse. It was more spacious than the cave of the Crib. Mary remained there some hours, while Joseph was making the latter more habitable. He brought also from the city many different little vessels and some dried fruits. Mary told him that the birth of the Child would arrive on the coming night. It was then nine months since her conception by the Holy Ghost. She begged him to do all in his power that they might receive as honorably as possible this Child promised by God, this Child supernaturally conceived; and she invited him to unite with her in prayer for those hard-hearted people who would afford Him no place of shelter. Joseph proposed to bring some pious women whom he knew in Bethlehem to her assistance; but Mary would not allow it, she declared that she had no need of anyone. It was five o'clock in the evening when Joseph brought Mary back again to the Crib Cave. He hung up several more lamps, and made a place under the shed before the door for the little she-ass, which came joyfully hurrying from the fields to meet them.
When Mary told Joseph that her time was drawing near and that he should now betake himself to prayer, he left her and turned toward his sleeping place to do her bidding. Before entering his little recess, he looked back once toward that part of the cave where Mary knelt upon her couch in prayer, her back to him, her face toward the east. He saw the cave filled with the light that streamed from Mary, for she was entirely enveloped as if by flames. It was as if he were, like Moses, looking into the burning bush. He sank prostrate to the ground in prayer, and looked not back again. The glory around Mary became brighter and brighter, the lamps that Joseph had lit were no longer to be seen. Mary knelt, her flowing white robe spread out before her. At the twelfth hour, her prayer became ecstatic, and I saw her raised so far above the ground that one could see it beneath her. Her hands were crossed upon her breast, and the light around her grew even more resplendent. I no longer saw the roof of the cave. Above Mary stretched a pathway of light up to Heaven, in which pathway it seemed as if one light came forth from another, as if one figure dissolved into another, and from these different spheres of light other heavenly figures issued. Mary continued in prayer, her eyes bent low upon the ground. At that moment she gave birth to the Infant Jesus. I saw Him like a tiny, shining Child, lying on the rug at her knees, and brighter far than all the other brilliancy. He seemed to grow before my eyes. But dazzled by the glittering and flashing of light, I know not whether I really saw that, or how I saw it. Even inanimate nature seemed stirred. The stones of the rocky floor and the walls of the cave were glimmering and sparkling, as if instinct with life.
Mary's ecstasy lasted some moments longer. Then I saw her spread a cover over the Child, but she did not yet take It up, nor even touch It. After a long time, I saw the Child stirring and heard It crying, and then only did Mary seem to recover full consciousness. She lifted the Child, along with the cover that she had thrown over It, to her breast and sat veiled, herself and Child quite enveloped. I think she was suckling It. I saw angels around her in human form prostrate on their faces. It may, perhaps, have been an hour after the birth when Mary called St. Joseph, who still lay prostrate in prayer. When he approached, he fell on his knees, his face to the ground, in a transport of joy, devotion, and humility. Mary again urged him to look upon the Sacred Gift from Heaven, and then did Joseph take the Child into his arms. And now the Blessed Virgin swathed the Child in red and over that in a white veil up as far as under the little arms, and the upper part of the body from the armpits to the head, she wrapped up in another piece of linen. She had only four swaddling cloths with her. She laid the Child in the Crib, which had been filled with rushes and fine moss over which was spread a cover that hung down at the sides. The Crib stood over the stone trough, and at this spot the ground stretched straight and level as far as the passage, where it made a broader flexure towards south. The floor of this part of the cave lay somewhat deeper than where the Child was born, and down to it steps had been formed in the earth. When Mary laid the Child in the Crib, both she and Joseph stood by It in tears, singing the praises of God.
Ultimately, as far as Catholics go, one can believe one way or another on this subject matter. As the Church has not pronounced definitely on this subject!