The people of Britain all spoke Celtic languages from which modern Welsh and Gaelic developed. Then they were invaded by the Angles, Saxons and Jutes and gradually pushed back to Scotland, Wales and Cornwall or assimilated into Anglo-Saxon (Jutish?) society, which spoke a language from which modern English developed. Seven hundred years ago the extremely nasty Edward I of England (who probably didn't speak any of the above-mentioned languages) conquered Wales and absorbed it into the administration of England; despite the huge pressure that puts on a culture and a language, Welsh has survived. At the same time Robert 1 of Scotland (Robert the Bruce) might have spoken Gaelic (his mother was a Gael) but the change away from Gaelic as the language of political power had begun there too. What we now call 'Scots' was then called 'Inglis'
The ancient Britons all spoke Celtic languages because they were all Celts. They lived all over the British Isles.
The Angles, Saxons and Jutes (Angles) invaded and they spoke a Germanic language.The Angles gradually pushed the Britons back to Wales, Cornwall and Scotland.
(Ignore the Vikings and the Normans for now because, although they had an influence, nobody actually adopted their languages.)
The Scots (who were from Ireland) settled in West Scotland, speaking a Gaelic form of Celtic.
The Gaelic Scots and the (possibly non-Gaelic but still Celtic-speaking) Picts later had some sort of political merger, which resulted in the formation of the Kingdom of Scotland, i.e. Kenneth MacAlpine, King of Scots became ruler of a united kingdom of Scotland. Strahclyde was separate and speaking a form closer to Welsh.
At the time of Malcolm Canmore's marriage to the Saxon princess St Margaret (she wasn't a saint at the time) the whole of Scotland would have been speaking Gaelic or some form of Celtic.
The very nasty Edward 1 of England conquered Wales and absorbed it into the administration of England, resulting in huge political and economic pressure on the survival of the Welsh language for the next 700 years.
Around the same time, Robert 1 of Scotland (Robert the Bruce) might have spoken Gaelic (his mother was a Gael) but the process of removing Gaelic from the court had begun. At that time, the Lothians were part of Scotland, but spoke 'Inglis', which is the Scots word for 'English' and is the dialect that we would now call 'Scots'. By the time of James IV (Battle of Flodden, 1513, commemorated this year) the Scottish court was speaking Inglis/Scots and the Gaels (i.e. the original Scots) were regarded as ungovernable savages who spoke Erse (Irish). Everbody knows about the prohibition of the kilt and the bagpipes after the "45, and the Highland Clearances, but it its little known that even in the twentieth century children were beaten for speaking Gaelic at school and the law stated that all lessons had to be conducted in English regardless of the first language of the children. (I don't know the situation in Wales in those days).
I kind of understand the views of people who do not wish to waste public money, but I think we have to see it in the context of the UK state's responsibility for almost obliterating some of our native languages and cultures. The value of another language is not just that you can use a few phrases when you go on holiday; it can represent a different way of thinking and that is very valuable. We spend a great deal of money trying to preserve variety in the natural environment for very good reason, and the same principles apply to culture and thought. Learning Gaelic/Welsh does not prevent anyone from learning French or German if they really want to study languages and it is difficult to see why the Gaels and Welsh should just give up on their languages and their culture altogether just to enable the rest of to save a few pounds.