I assume you mean King Herod, not King Harold! ? The oldest manuscript of the Bhagavad Gita is from about 1490 AD. It is in the Bodelian Library at Cambridge University. This is many centuries after Christ. Having said that, the original poem comes from well before the fifteenth century AD. When was it composed? A better question, if one wants to support the (to me) preposterous idea that the New Testament writers used the Bhagavad Gita to invent the story of Herod, is when was the Bhagavad Gita written down, and when might actual copies of the document have found their way to Palestine?
Based on linguistic studies, my research tells me that the original was composed sometime in the third century AD or later. In other words, in the 200s BC or perhaps the 100s BC. So, it probably existed a hundred years of so before Christ, but it is difficult to prove. However, quite likely it only existed in oral form. When it was actually written down is not known. Perhaps a few hundred years AD. When might it have gotten to Palestine? Who knows? Hundreds of years later? Is there ANY evidence if the influence of Hinduism on the Jews in the first century? NO! None! Certainly not the Bhagavad Gita which most likely was not even written down until after Christ. To be honest, the idea that a group of Jewish followers of a Jewish Rabbi/teacher Jesus would incorporate something from Hinduism is preposterous.
Besides, in order to do this, they would have to tell a lie about Jesus. They would have to tell a lie about Herod who had lived within the lifetime of many of those who heard Jesus. Again, can any rational person actually believe such a proposal? The proposal is this: The Jewish followers of the Jewish Jesus, who had worked literally thousands of miracles want to get people to believe that Jesus is the Messiah. So, they start reading the Bhagavad Gita, they find a story there, and then they lift that story from a Hindu myth, and pretend that this actually happened to Jesus. They told this to all the disciples of Jesus and convinced them that it true, despite the fact that some of those who would have known whether it is true or not (such as Mary!!!) would obviously be able to correct the lie.
Really, this is SO preposterous that anyone proposing this idea ought to be truly embarrassed of themselves! How gullible do we need to be to accept this crazy idea? This did not happen. End of story, and anyone who tries to tell you so ought to be chastened for proposing such an outlandish idea.
By the way, to add to the levels of absurdity to this story, have you actually read the Bhagavadh Gita and what it actually says about Putana and Krishna? The best I can tell, even if we allow the preposterous idea that the apostles would read the Hindu scripture and incorporate part of it into the gospel, when I look at the story, I see basically no parallel between the stories at all. My research tells me that in the myth Putana was killed by the infant “god” Krishna. I do not see a king, jealous about a god-man, having children killed in order to destroy Krishna. In the Bhagavata Purana (not the Bhaghavad Gita, and written many years after the New Testament) there is the story that Putana was sent by Krishna’s evil uncle to kill him. How is that like the Jesus story? This entire idea is a complete fabrication, the best I can tell. Can you point me to where in the Bhagavad Gita this supposedly parallel story is found? I could not find it.
One more thing. You ask “or the other way round.” I say a hard no to that proposal as well. It is likely (though not proven) that the Bhagavad Gita was first composed a hundred years or so before Christ. Besides, as unlikely as it is that a Hindu myth would be incorporated into the Christian gospel, it is at least as unlikely that a true story from a relatively insignificant Jewish people from 2000 miles away would be incorporated into a Hindu myth. Besides, the story of Herod and Jesus has little if any real parallel to the story of Putana and Krishna. No. I say no!