In fact, the historical consensus these days is that the majority of England's native population did not move westward. There certainly was significant migration to Brittany, but this appears to have mostly come from Dumnonia, which the Anglo-Saxons didn't take over. There's an awful lot I could write about this, but that isn't really your question.

'Celtic' is essentially a term the Greeks and Romans used to describe a wide range of peoples who had closely related languages and shared cultural roots. These peoples stretched from Ireland to Turkey, though Celtic culture appears to have its origins in central Europe. To what extent the culture was spread by migration and how far by trade is a matter of debate.

'Celtic' can be a useful term for describing the culture and language of peoples such as the Welsh, Gaels and Bretons. But it wasn't a word these peoples used to describe themselves. The Britons called themselves pretanni in Latin or brythoniad in Welsh, both terms meaning 'Britons'. By the tenth century the Welsh had come to call themselves cymry, which essentially means 'fellow-countrymen'. They call themselves this to this day. However, the name may be even older, since St Patrick, a Briton writing in the fifth century, used the term 'fellow-citizens'. Cives is also used by Gildas, and later Brittonic Latin writers.

'Welsh' comes from a Germanic word that applied to the inhabitants of the Roman Empire, and usually to Romanised Celts. Forms of this word can also be seen in 'Walloon' and 'Wallachia'. Today, German Swiss people sometimes call French and Italian Swiss people welsch. In Old English, the word is first attested in the laws of Ine of Wessex. The laws show that 'Welsh' people lived among the West Saxons and that some held considerable wealth. So the term probably meant those who primarily spoke Brittonic or Latin, rather than the people of a separate country.

However, the Anglo-Saxons did also call the Welsh 'Britons', particularly in Latin texts such as Bede's Historia Ecclesiastica. 'Briton' and 'Welshman' seem to have basically been synonyms, the former being the Latin term and the latter being the Germanic one. The Gaelic rulers of Scotland also came to use the term 'Britons', when referring to the Britonnic people of south-west Scotland and Cumbria.

In summary, the Britons were Celts. However, they didn't conceive of themselves as such, because no one except possibly a few monks had ever come across the word 'Celt'. They didn't have a sense of solidarity with other Celtic peoples such as the Irish, or even the Picts (who were also a Brittonic people). Our sources suggest that they were considered as foreign as the Saxons were. The Welsh of Wales considered themselves cousins to the Cornish and Bretons, but because they were also Britons, not because they were Celts.