It had been so many centuries since the Celts arrived in Britain that I assume they Britons themselves would not have known about the relationship. But the two groups certainly may have shared many traits that were evident to outsiders such the Romans.
The first difficulty arises from the term ‘Celts’. What do we mean by ‘Celts’? Until the late-twentieth century, the general view was that Celtic culture was spread to Britain by invasions in the Iron Age. The different ‘Celtic’ cultural phases e.g. the Hallstatt Culture and the La Tene Culture, were understood as being evidence of the arrival of these invaders. However, the distribution of ‘Celtic’ finds suggests a more gradual diffusion of Hallstatt and La Tene elements from the Continent; you’re much more likely to find an archetypically ‘Celtic’ artefact in South-East England than in Ireland. Moreover, the limited genetic evidence from excavated skeletons indicates no great genetic difference between Bronze Age and Iron Age Britons. Current scholarship usually regards the spread of the Hallstatt Culture to Britain as taking place largely through peaceful trading contact. The idea that the archaeologically identifiable spread of the Hallstatt Culture can be viewed as the spread of Celtic culture is not, however, uncontroversial. Barry Cunliffe proposes the Atlantic seaboard as the birthplace of Celtic language and culture, while Patrick Sims-Williams theorises an origin in central Gaul. Indeed, a lot of Celticists believe that Celtic language may have spread during the Bronze Age. Although we can say when certain archetypically ‘Celtic’ artefacts and cultural practices came to Britain, we can’t say for sure when Britain actually began to use Celtic languages and practise Celtic culture.
If Celtic language was long-established in Britain by the first century BC, what reason would the Britons and Irish have to consider themselves ‘Celts’? After all, they had very little contact with non-Celtic peoples. There is no evidence of a ‘Celtic’ identity existing in Britain or Ireland before modern revivalist movements. The earliest writers from Britain and Ireland; Pelagius, Patrick, Gildas, Columbanus; make no mention of ‘Celts’. Indeed, for Gildas and later Welsh writers, the Irish were just as foreign as the non-Celtic Anglo-Saxons. This is to say that there is no evidence for any widespread concept of ‘Celticness’ among the Britons themselves.
But not only do the Britons not seem to have thought of themselves as Celts; it would appear that the Romans did not think of them as Celts either. There are plenty of instances where Roman writers call Gauls Celts, but not a single one where Britons are referred to as such. Zosimus’s Historia Nova, a source that frequently uses the term ‘Celtae’ explicitly refers to the Britons separately from the Celts. It would therefore appear that the Romans did not regard the Britons and Gauls as being part of a single Celtic ethnic group.
This is not to say that the Romans viewed the Britons as unrelated to the Gauls. Julius Caesar tells us:
The inland part of Britain is inhabited by tribes declared in their own tradition to be indigenous to the island, the maritime part by tribes that migrated at an earlier time from Belgium to seek booty by invasion. Nearly all of these latter are called after the names of the states from which they sprang when they went to Britain; and after the invasion they abode there and began to till the fields. The population is innumerable; the farm-buildings are found very close together, being very like those of the Gauls … Of all the Britons the inhabitants of Kent, an entirely maritime district, are by far the most civilised, differing but little from the Gallic manner of life.
Tacitus, writing after the conquest of Britain, tells us:
They who are nearest Gaul resemble the inhabitants of that country; whether from the duration of hereditary influence, or whether it be that when lands jut forward in opposite directions, climate gives the same condition of body to the inhabitants of both. On a general survey, however, it appears probable that the Gauls originally took possession of the neighbouring coast. The sacred rites and superstitions of these people are discernible among the Britons. The languages of the two nations do not greatly differ. The same audacity in provoking danger, and irresolution in facing it when present, is observable in both.