Let’s look at the languages of Scotland as an example (I am a linguist after all!).
Today the main language of Scotland clearly is English (or rather Scottish Standard English) — everybody knows it, and it’s the main language of government, education, etc.
However, according to the latest census, 1.5m people (out of 5m) also speak Scots, a closely related language (the distance between Scots and English is a bit like Danish and Swedish). However, it hasn’t got much support in the education system and it’s often stigmatised. Both English and Scots are of course derived from the language of the Anglo-Saxons, who immigrated to Great Britain from what is now Denmark and Germany more than 1500 years ago.
The third of the extant languages is Gaelic, which has only got 50k speakers left. Many people seem to think that it’s Scotland’s original language, but this is only really true in the sense that it was the dominant language at the time of the unification of Scotland. Gaelic is also an immigrant language and has its roots in Ireland.
For centuries, another Scottish language was Norn, the descendant of Old Norse, the language of the Vikings. It was spoken in large parts of Highlands and Islands for a while, but for the last few centuries of its existence it was confined to Orkney and Shetland. The last speaker died around 1850.
If we go further back, the language of large parts of southern Scotland was Cumbric (or Brittonic, Brythonic or Old Welsh), which was closely related to modern Welsh. Indeed, Welsh speakers still know legends set here in Scotland. It was described well on Wings over Scotland recently:
[Visiting Scotland by train] also takes us through the lands of “Yr Hen Ogledd” (the old north), the heartland of the old Brythonic language, the prototype of modern Welsh and the seven kingdoms which established themselves in the intervallum of several centuries after the Romans left these shores in 400 AD.
The old Brythonic names of these kingdoms such as Ystrad Clud (Strathclyde), Galwyddel, (Galloway), Aeron (Ayrshire) and Lleddeiniawn (Lothian), are instantly recognizable to a modern-day Welsh speaker, and being confronted with a cultural link which stretches back over well over 1,000 years cannot fail to touch one deeply.
Finally, the language of the northern two-thirds of Scotland used to be Pictish, which was probably related to Cumbric (although we don’t know for sure).
The historical records don’t allow us to go back any further, but Cumbric and probably Pictish were Celtic languages, and Celtic is a branch of the Indo-European language family, which means that these languages too must have been immigrant languages at some point (the Indo-European languages probably originated near the Black Sea). Alas, we don’t know which language(s) they replaced.
All we know is that Scotland has always been multilingual.