Any idea of the date for the Abraham nativity tradition?
I think it's late. So I doubt one could rule out Christian influence, though the stories are quite different despite the similarities. The story that I quoted seems to go back to a 12th century midrash (Sefer ha-Yashar), and the Jewish Encyclopedia assures us that it goes back to "the Palestinian rabbis of the second century, and afterward further developed under the influence of Babylonian folk-lore," but I can't find intermediate versions of the story at short notice. I know however that the connection with the star and astrologers is suggested by Abraham's birthplace as "Ur of the Chaldeans" (cf. "Chaldeans" in Daniel), and Genesis Rabba, Jubilees, Philo, Pseudo-Eupolemus, Artapanus and other sources develop this theme of Abraham's knowledge of the stars. For instance, Pseudo-Orphica (B), cited by Clement of Alexandria, referred to Abraham as "a certain unique man, by descent an offshoot of the Chaldeans, for he was knowledgeable about the path of the Star, and the movements of the spheres around the earth" (Clement of Alexandria, Miscellanies 124).
The only other text I could find that refers to the story is the 16th-century Cuando el Rey Nimrod, a Ladino folk song of the Sephardic Jews in Spain, which runs as follows:
"When King Nimrod went into the fields he looked at the heavens and at all the stars, He saw a holy light above the Jewish quarter that was a sign that Abraham our father was about to be born. Immediately the midwives recommended that every pregnant women tarry because if a son were born they would have to kill him. (That told of the birth of Abraham our father.) Terah's wife was pregnant. Daily he asked her the question: 'Why is your face so pale?' Already she knew the good she had within her at the end of nine months she was determined to give birth. She walked through the countryside and vineyards. Her husband did not know she was gone. She found a cave in which to give birth. At that time the newborn spoke: 'Go out of the cave, mother. I have already found someone who will remove me. An angel from heaven will accompany me because I was created blessed from heaven'. After twenty days she went to visit him. She saw in front of her a young man leaping, looking at the sky, aiming to understand, in order to know the God of truth. 'Mother, my mother what are you looking for here?' 'I gave birth to a precious son here. I came to look for him here. If he is alive I will be consoled.' 'Mother, my mother, what are you saying? How could you leave your precious son? After twenty days how do you visit him? I am your precious son. Look mother, God is one, the creation of the heavens was one by one. Tell Nimrod that he has lost his common sense because he does not want to believe in the True One. King Nimrod did not manage to learn. 'Bring him here immediately, before they cause a rebellion and encourage others to believe in the True One And not in me.' They brought him in great humility. He strongly grabbed Nimrod's throne. 'Tell me, evil one. Why do you think you are God. Because you do not want to believe in the True One.' 'Light a fiery furnace, throw him in immediately, protect yourself from him because he is sharp. If God allows him to escape, then He is real' " (Cuando el Rey Nimrod; listen to it HERE).
This version is different from the one quoted earlier (e.g. no astrologers, no gifts given to Terah, etc.), but there is still the juxtaposition of the star motif with the massacre motif. Giving birth in a cave is reminiscent of Mithra mysteries and certain apocryphal Jesus traditions. The passing-through-fire motif towards the end of the song is indeed very old, found throughout Jewish midrash on Abraham (cf. Targum Neophyti Gen. 11:31; 15:7), Genesis Rabba 38:13, Pseudo-Philo, Antiquities 6:4-5; 23:5), and suggested by the name "Ur" as the city Abraham was born. The idea of Abraham's heresy of worshipping the "True One" also goes back to Josephus, Antiquities 1:154-157, who claimed that while still in Ur, Abraham "became the first person to argue that there is a single God who is the creator of things....because of these ideas the Chaldeans and the other people of Mesopotamia rose up against him".
I'll try to find other possible versions of this story....