The language of the Rheged ruling classes, and quite likely of the majority of the population, was a southern Celtic language somewhat similar to Welsh.
The language of Northumbria was Old English, and it is likely there was some English presence in Rheged even while it was still independant.
There is considerable evidence for an Irish presence as well. It is known that Irish [Christian missionaries]? were active in Rheged (although the region was at least nominally Christian even in Roman times) and there were likely Irish traders, pirates and settlers unconnected with the church as well.
The poet Taleisin? lived in Rheged.
One of the more accessible sources for Rheged history (and dark ages British history generally) is the works of the Venerable Bede.