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 independent.
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.
One of the more accessible sources for Rheged history (and dark ages British history generally) is the works of the Venerable Bede.
The name Rheged has been adopted by the Rheged Centre close to Penrith, Cumberland. The centre apart from having a number of retail outlets boasts the largest turf roof in Europe and a giant cinema screen which shows among other topics a film about the history of Rheged.