Welsh does not mean Roman. It’s the Anglo-Saxon term for people unlike themselves, deriving from the wealas or “foreigners”, and in those times it also had derogatory connotations. The Germanic Franks, who spoke a language not too different from Old English, applied a similar term to the Latin-speaking/Romanised Gauls they encountered in more or less what is modern Wallonia and the term Walloon (Flemish/Dutch: Waals), for a French-speaking Belgian, is still used today. The Balkan term Vlach also has a similar etymology, passing from Old German into Slavic languages.
The nomenclature in terms of Cornwall is also complicated, because it was also Dumnonia and yet became known as the land of the Cornubian/Cornovian Welsh and yet again it was also referred to as West Wales. In Britanny, there is an area which is also called Cornwall (Kernev/Cornouaille) to this day and, historically, there was also a Domnonée amoricaine. Brittany itself was Armorica – ar mor – the land by the sea and became Brittany – Little Britain – owing to power shifts and colonisation of the sub-Roman period. There was also a relatively short-lived Brythonic enclave in northern Spain, Britonia or Bretoña, in what is now Galicia.
The Romans conquered what we know as Wales, finally, under Suetonius Paulinus in about 60 CE.
Urien Rheged was not a “Welsh King” as in what we call Wales (in English) today, he was actually a king of the Hen Ogledd, or the Old North, which was an area of Northern England and the Lowlands of Scotland. The language that Taliesin would have spoken would be more accurately described as Cumbric or a northern dialect of Brythonic, albeit much akin to Old Welsh. Remember that the Welsh word for the Welsh is Cymry and comes from *combrogi, the “compatriots” or “companions”, it doesn’t refer to a geographic entity, but rather to a national self-perception. During the early periods of Welsh literature, the word Britons was still being used. We need to take great care using terms like England, Scotland, Wales or Cornwall during that period as we run the risk of anachronisms and an inaccurate representation of the geopolitical and cultural dynamics of the areas in question.
If you’re interested in non/pre-Christian survivals in this period, you should research the rather enigmatic figure Myrddin Wyllt, who may well be the same as the figure that has come down to us as Lailoken and contributed to the later composite character of Merlin.