Latin would have been spoken amongst the monastic communities and would have been the primary language of communication between Britons (the Welsh, Cumbrians and Cornish) and the Anglo-Saxons.

Latin as a primary language is an interesting thought, and I have to admit I don't think that the Venerable Bede had that in mind. There are three possibilities I can think of.

First, Bede could have been referring to his colleagues and brothers, many of whom would have learned Latin in order to read, write and pray. Most international communication, especially beyond the archipelago, would have been in Latin. To the best of my knowledge that remained the case until the era of the Crusades.

Second, he is referring to the language of the sub-Roman continental traders. There was trade between the Anglo-Saxons and Aquitaine, Italy and the Greek East. Merchants from the first two would have spoken a language far closer to Latin than Italian or Occitan are today, and a trader from the East could have learned Latin any number of ways.

Third, there might still have been Latin speaking communities in Northumbria, in the unsettled regions between Strathclyde, Pictland/Scotia, and the Anglo-Saxon kingdoms. That seems incredibly unlikely and is literally the only scenario I can think of to make your initial question possible. A century or so earlier than Bede it may have been possible to meet people whose birth language was Latin in Devon, but that would be it.

The vast majority of the assimilated Romano-British lived along the South-East of modern England, with further outposts around York and up north along Hadrian's wall. Not coincidentally, these were the lands that the Saxons conquered first and fastest.

Many of the more Romanized Britons didn't flee to their countrymen in the west of the island, they fled south, to modern Brittany and Normandy. I'm not sure what they thought when they escaped the Saxons and then the Franks turned up, but I'm sure it was hilarious.

Much of Wales was never settled fully by Roman colonists, and the local relation with the colonial administration in York or London was one of grudging tax-payment. Roman norms perhaps continued longer in Cornwall and Devon, but the majority of the Romanized population seem to have left in the fifth and sixth centuries following military setbacks. Were there some families with Latinate names? We know there were, but there are names with Greek origins in families that haven't spoken Greek since the fall of Bactria.

As with so much else from this period, linguistic history is hard to trace, but the campaigns, wars and population movements are relatively well attested. I would recommend 'Arthur and the Anglo-Saxon Wars', (if my memory serves me), from Osprey, as a fairly simple study of the period.