India is well outside the bounds of Roman influence, but there is evidence of Christianity in India very early. After the faith reached India, how much contact was there with Christianity in the Mediterranean world for the first 6 or 7 centuries?
The Christian's in the West did have some loose connection with those Christians in the far east due to the trade that went on with the Silk and Spice trail. The Christians of Far Asia and The Roman empire however were mostly separated due to the political divide between Rome and the Parthian empire, and of course the vast distances that existed at a time before modern communication inventions were developed. (I will point out an Illustration of this, the Christians of the Syriac, "Church of the East" that lived in Iraq and further East accepted the first two Ecumenical Counsels that formed the early Nicene Creed but did so 40 years after the fact, and due to the distance and traveling difficulties was not able to send delegates to both those counsels!)
Early Church historians like Eusebius, report the account that the apostle Thomas brought the gospel to India (around AD 52 according to Wikipedia) to The Malabar coast, saint Thomas was said to have been martyred by a Hindu, who stabbed him in a cave with a spear. Besides the Thomas tradition, Christians of the East Syriac (Nestorian) tradition also gradually moved into the area as well sometime after the Fourth Century. But needless to say, there has been a small Christian presence in India since the time of the writing of the New Testament, and for some centuries it was quite significant in numbers and social status (Christians in medieval India had a caste rank that was right below that of the Brahmin, the top caste of India).
If you are interested in this Topic, I recommend, "East of the Euphrates", by TV Philip a member of the Mar Thoma Church of India. See the amazon link here https://www.amazon.com/East-Euphrates-Early-Christianity-Asia/product-reviews/8172144415
(PS - I was once a member of the a Church of the East group, that had missionary ties to India.)