You're going to have to be more specific. Do you mean early church as in St. Thomas? If so, there would be no need to communicate with Rome. They weren't Roman Catholics and the doctrine of Papal primacy hadn't been fully developed yet.

If you mean the coming of the Portuguese, they would've worked through a nuncio (i.e. Francis Xavier) who would represent the Holy See. He would send letters back to Rome for the Pope and for the Superior General of the Jesuits in the case of Xavier. The average transit time would've been around 4-6 months. So there was a serious delay, but the impact on the mission would've been minimal. Xavier would've been able to make most decisions in his capacity as nuncio and if a major doctrinal issue came up (none did) they could've waited for a response from the Pope.

Source William V. Bangert, S.J. A History of the Society of Jesus (1986).