Historically, the view on colonialism and imperialism inside Britain has been affected by our own empire. It is not a coincidence that Britain begins to be interested in the idea of Romanisation i.e. becoming Roman in the Jacobean period, around the settlement of Jamestown, and that by the 1800s the view of Roman Britain is essentially that of contemporary India - elite Romans living in fortified complexes and villas while outside indigenous workers live in their own villages or labour as slaves. There has always been an identification inside Britain of the Roman Empire with the British Empire, and a subsequent ripple of concern - after all, the Roman Empire fell, and why shouldn't the British also? For a particularly blunt take on this, I would recommend the introduction in the first edition of Collingwood Bruce's Handbook to the Roman Wall. In the last thirty years or so, the concept of Romanisation has collapsed as we've started to think about it in a post-colonial setting and to look for more complex histories of resistance and cultural assimilation.

There are no surviving sources that directly give their thoughts on the conquest or occupation (Tacitus' famous line given to Calgacus that they plunder, they slaughter, they steal, and this they falsely name empire; they make a wasteland, and they call it peace is, while an extremely powerful inditement of Empire, ultimately his own work and possibly more about his own feelings about the Emperor). Although some writing survives, it is largely in the form of small inscriptions such as curse tablets found in baths, or accidental preservation such as the Vindolanda letters, all of which represent some level of indentification with/assimilation to Roman culture. Even where non-Romanised names exist on these inscriptions, the fact that they are being created at all - and written in Latin - demonstrates some identification with Roman culture. I think we have maybe one inscription from the Bath curse tablets that is definitively Old Brittonic, and no surviving documents.

This makes it impossible to say exactly what they thought or felt, or how they expressed themselves. There were, of course, many cultures inside Britain, all of whom would have had a different experience of becoming and being part of the empire, and these experiences would have been further mediated by class/gender/etc. We can certainly say that they had a variety of responses to Rome and to Roman power. In terms of elites, some people such as Caratacus (who again gets a lovely speech from Tacitus about resistance against slavery!) obviously did resist in violent ways, while others like Cartimandua chose to instead work with the Romans. Some like the Iceni royal family apparently chose first one and then the other as circumstances dictated. However, again we only have other's words for what they were thinking and experiencing; how they themselves would have conceptualised this is unknowable.

There is some room to talk about material culture, which tells us more about what ordinary people were doing and thinking, but this is a difficult topic; trying to track cultural identity through objects leads to a lot of jokes about the Swedish Empire that expanded across the world in the 1970s and 80s. Just because a lot of people have an ikea sofa doesn't mean we all identify with Swedish culture! We can look at how architecture changed - in the first and second centuries there are a lot of Roman-style buildings created in urban centres, which suggests that at least there is an attempt being made to bring Roman-style living and communal organisation into Britain - and at the continuity of burial practices and foodways which persist for some time. When we look at religious practices, we can see local sites and deities - and even some practices - persist, even as they're identified with and folded safely into a Roman pantheon. Aqua Sulis is a particularly good example of this; we have an older god, Sulis, who becomes identified with Minerva and worshipped as Sulis Minerva.

There's an interesting argument to be made here about the end of the empire in Britain; many civic projects and parts of town centres are abandoned from the 350s onwards, not unlike the way in which parts of Detroit became disused and no longer viable, and it could reflect a return to indigenous style architecture (especially wooden architecture, which is harder to see archaeologically) and ways of life. That, of course, suggests that the province was never fully culturally "Roman", whatever that meant. Certainly the north was "less" Roman than the south, and more heavily militarised. Some writers have taken the argument further - that as Britain has a bad habit of electing new western emperors for itself towards the end of the period, that Rome did not so much abandon Britain as unviable as was thrown out by local elites. I personally find it a little unsupported by any real evidence, but it's an interesting bit of speculation. The abandonment of many civic buildings and centres could just as easily point to economic distress - pre-modern cities needed an influx of people to them as they were not self-sustaining, and if there is no longer an important reason for people to move to them then they died relatively easily - and an elite which is no longer wealthy or interested enough to invest in those specific community projects, especially as specialist activities such as butchery seem to continue there for some time, often moving into those civic buildings.

This is maybe less helpful then you were looking for, but in conclusion: it is absolutely unknowable. No one who was indigenous to Britain wrote down what they thought about it and had the writing survive, and interpreting material culture and identity is extremely difficult. I'd recommend for further research starting with Richard Hingley's work, as he's extremely good on post-colonial takes on Roman Britain, and going outwards from there.