I think you really have two different though closely related questions here. The first is how have people communicated with peoples they have no previous knowledge of or common language with. The second is how have different polities (the term nation has implications I won't get into here) establish communications with each other.

I will be answering the first question but in doing so may reveal insights about the second, as communications between polities is necessarily done through people. On the linguistic side of things one should note that people can learn languages they have no experience of if immersed in it. I'll use the example of the Spanish first arriving in Mesoamerica as I've read Bernal Diaz's account of the conquest of Mexico recently enough that its still somewhat fresh in my head.

Diaz was part of the first expeditions from the Spanish colonies in the Caribbean to what is today Mexico. On the first journey the party lands in the Yucatan peninsula and are ambushed by the local Maya people. On retreating the Spaniards manage to capture two locals who learned Spanish over time and acted as interpreters between the Spanish and the Maya leaders they encountered.

When the Spaniards returned to Mexico on the voyage with Hernan Cortes at the helm that would become the voyage of the conquest of Mexico they land on the island of Cozumel off the Yucatan coast and hear that there are Spaniards living as slaves on the mainland. Diaz recounts that these two Spaniards were survivors of a shipwreck in the Caribbean and inadvertently become the first Spaniards in Mexico. They were enslaved by the Maya and learned their language. One of the men, Gonzalo Guerrero, became part of the community and married a local woman, whereas the other, Jeronimo de Aguilar remained a slave. When the Spanish party arrived Guerrero chose to remain with his Maya family and community while Aguilar joined the Spanish fleet and acted as a key interpreter between the local people and Spaniards.

After spending time in and around the Yucatan the fleet moves westwards towards Tenochtitlan and the Aztecs. Aguilar does not speak the languages of the peoples of central Mexico but following another battle on the Yucatan coast the Spanish are given a number of slaves, including a woman known as Marina or La Malinche. La Malinche serves a key role in communication between the Spaniards the people of central Mexico because despite not knowing Spanish she does speak one of the languages that Aguilar also speaks (I believe it was Chontal Maya though I could be mistaken). La Malinche worked with Aguilar as interpreters for Cortes and the rest of the fleet, at least until she learned enough Spanish translate on her own.

So by the time the Spanish arrived in the Aztec capital they already had an interpreter capable of speaking the languages of both parties, enabling them to communicate with each other.

One should note that this is but one primary source of the conquest of Mexico and Diaz was not merely an observer but active participant in these events. He has a tendency to focus on the actions of his fellow Spaniards to a greater degree than that of others such as the indigenous peoples involved. While there are a number of other primary sources related to the conquest of Mexico I can't speak to the extent to which they mention communication barriers or translators.