I'm no expert on Britain under so feel free to correct any assumptions I make the are way off.
An element of British society became Romanised whilst it was a province of Rome. Their soldiers fought abroad, they paid taxes, they participated in the Roman economy. Villas and urbanisation were widespread. Once the Roman rule collapsed there was attempts to hold together this Romano-British society that ultimately failed.
However what I've never been clear on is whether there was ever a discrete frontier between "Roman" areas and "Briton/Celtic" areas. The Iceni were made into a tributary civitas and had an established capital- yet they don't seem Romanised. However Londinium was a major Roman city with forum, temples, legionary fort. Who lived there? People who abandoned their tribes for civilisation? Roman immigrants?
Another complicating factor is that there are Roman villas throughout England, including on land supposedly part of the Iceni's territory. So were they Iceni who decided to go full-Roman and build themselves a villa, or Romans who somehow took their land? Did they pay rent, or was it won in conquest, or what?
Did the tribes live and just avoid getting in the Roman's way when they wanted a road or city in their territory or was there some arbitration process?
They did not supplant local culture nor did they uproot them. The Romans had no policy of deliberate cultural replacement and it is easy to see many ways (such as continued ritual activity at megalithic sites) that Britain and the other areas of the Roman empire remained distinctly themselves. There was no homogenization, in short.
This is not to say, however, that nothing changed. In a way, everything changed. While the Romans did not institute cultural policies, the so-called experience of empire affected everyone. This isn't just a matter of elites across the entire territory changing Iron Age style dwellings for villas (although I personally think this signals a major change in lifeways and social relations), the changes go down to the merest pot, as ceramic production structures and techniques became "Romanized" and even very humble dwellings used ceramic rooftiles. The development of cities also cannot be viewed as merely urban phenomenon, as they affected rural areas profoundly. Not only because they represent concentrations of rural produce (ie, the grain needed to feed the city) and rural immigration, but also because they represent nodes of economic production and exchange. The mere act of, say, selling your grain at market was very different, not to mention the changed structures of taxation of resource extraction. It is important to remember that
This pattern holds more or less to the borders of Saxon Northumbria (it is worth noting, incidentally, that as much as the Iceni revolt has been talked about, it occurred roughly one decade after conquest and Roman Britain survived for over three and a half centuries after, so it is best not to make too much of a big deal out of it). Between that and the Wall Roman patterns of life did not really take hold. This is generally believed to be because the patterns of social control in that are pre-conquest were not easily exploited by the Roman conquerors. In southern England, where kings and other elite had substantial control over their territory Roman methods of co-option could easily maintain control. In northern England, where the elites had far less authority, they could not.
This is an extremely studied topic and I am trying to avoid the thorny question of what Romanization means and whether the Britons "felt Roman", but I feel from a material standpoint this is fairly uncontroversial.