Old Norse and Old English
It's basically how languages spread in pre-modern times. Have you seen "Vikings"? There is a scene in the first season when Ragnar lands in England and meets a local landlord and his soldiers on the beach. They're not able to communicate immediately. The vikings speak Old Norse, the English - well, Old English. It's exactly the situation you described: these are both languages coming form the common proto-Germanic roots, but they are separated by hundreds or maybe a thousand years of development. The only difference from what you wrote is that no language stays the same - the "original" language of people who were left behind in Scandinavia also evolved in time.
I wrote an article about it for a Polish linguistic website woofla.pl, but it's in Polish. All I have in English is an animation showing the development of Germanic languages: https://www.youtube.com/watch?v=UzYjsuvHx4c
In short, Germanic people settled in about 2000-1500BCE in the area of the modern Denmark and southern Sweden, and lived there in relative isolation until around 500-200BCE. The isolation contributed to the differences between Germanic and all other Indo-European families of languages. At the end of that period a dialectal continuum emerged with people who dwelt in southern and western parts on one end of the spectrum, and population in the Scandinavia on the other. From the first group, Western Germanic languages emerged, from the second, Old Norse.
Western Germanic languages were also a continuum for a long time (even until the modern times to be precise - even in the 18th century the border between Dutch and Low German was kinda fuzzy). In the south, close to the Alps, Old High German started to develop. In the north, Low German dialects. One of them was Saxon which was probably close or the same as the languages of Angles, and together these two peoples invaded Britain in the 5th-6th century CE. Old English developed as a merger of Old Saxon, Old Frisian, and some other, smaller dialects of people who joined the invaders. Some of them might have even spoken the northern dialects.
In the same time Old Norse was developing as well, but in a more conservative fashion. In the times of Ragnar, late 8th and early 9th century, it was still one language with just some differences between east and west Scandinavia. Even now Danish, Swedish and Norwegian (which has its own interesting story and is divided into two main "versions", Bokmal and Nynorsk) are quite intelligible.
Anyway, when the vikings invaded England it was as if the linguistic history made a full circle. And now both peoples needed translators. But not for long: the languages were still similar enough that probably after a few years of learning a Dane was able to speak Old English quite well, and vice versa. Only the later Norman invasion introduced a lot of (Norman) French vocabulary and put English on a very different track.