The Roman province of Britannia in the 5th century was in the middle of a horrific economic collapse that marred the northern periphery of the Roman world. Robin Fleming's Britain After Rome deals with this economic collapse pretty extensively and that's the source I'll be using for the majority of this answer. The evidence for this economic collapse is pretty clear, imports and exports from the rest of the empire dramatically ceased and Britain lacked the means to make up for this decline.

Britain had been a part of the Roman world for centuries by this point, but it did have the reputation as a backwater. It participated in a couple rebellions, propelling Cnstantine's father into the Imperial limelight for example, and a couple of religious disputes, and was unsecure enough that there were permanent stations for soldiers to fend off raids from the sea and land.

The big question about culture in Britain at this time is whether the population had Romanized or was still "Celtic". We know that there was a Romanized urban society in Britain at the time, but that had begun to decline rapidly with the economic downturn of Late Antiquity. It was also focused in urban centers on the east coast, and villas through the center of the countryside. Lack of documentation form Britain complicates our understanding of how ingrained Roman civilization had become in Britain however. For example its not clear how deep Latin had penetrated into the society and whether the "average" dweller still spoke a Celtic language or had started to assimilate into a nascent Romance language. We know that the island had been Christianized during Roman rule, and Christian communities would survive the Anglo-Saxon migration. However Christianity would come to be supplanted by the paganism brought by the Germanic migrants across much of what is today England.

These Romano-British populations did not totally disappear in Britain of course, they not only gave rise to the Welsh kingdoms of the early Middle Ages, but also survived to a limited degree under the regimes that the new Germanic migrants dominated, although many in time would assimilate to the new Anglo-Saxon identity that was cropping up across the lowland parts of Britain.

If you are looking for more specific aspects of British society I'm happy to try and field more specific inquiries, but keep in mind the limited surviving documentation, and spotty archaeological record for much of this time, may make some firm answers all but impossible.