I'll take a stab at medieval English views on the relationship between the Welsh and the ancient Britons.

Geoffrey of Monmouth, in his mythic history of Britain's past, presented the Welsh as the last holdouts of the ancient British people:

[the Saxons] landed in parts of Northumbria and occupied the waste lands from Albany to Cornwall. There was no inhabitant left alive to stop them, except for a few little pockets of Britons who had stayed behind, living precariously in Wales, in the remote recesses of the woods... As the foreign element around them became more and more powerful, they were given the name of Welsh instead of Britons: this word deriving either from their leader Gualo, or their queen Galaes, or else from their being so barbarous.

As you can tell from his name, he was a Welshman himself (though one who gives us a mixture of very positive and very negative remarks about his countrymen). However, his work was quickly translated into English and became central to the Matter of Britain. Even before this, the Welsh were seen as the descendants of the ancient Britons, though rather fallen from their glory days. Bryttas and Brytlande were used synonymously with W(e)alas in 11th century entries in the Anglo-Saxon Chronicle. In the same century, the authors of the Encomium Emmae Reginae and Vita Aedwardi Regis were content to refer to Wales as 'Britannia' and Gruffudd ap Llywelyn as king of the 'western Britons'.

Professor Huw Pryce wrote a paper describing how the terminology for Wales and the Welsh seemed to change over time. The 9th-12th century Welsh latin authors referred to themselves as Britones and their land as Britannia, but from the 12th century they seem to have accepted the Old English-derived Walenses, which meant 'foreigner' in OE, as a name for themselves and Wallia as a name for their country; Owain Gwynedd styled himself rex Wallie. In the Welsh language itself, the term Combrec seems to have been used in both senses for 'British' and 'Welsh'. English sources noted this usage and a prophetic tradition that a deliverer would come to unite the Britons and expel the English from the island. For example, Archbishop of Canterbury Hubert Walter complained to the Pope about the notion of a Welsh claim on the whole island in 1199. While scribes at Canterbury and royal charters had ceased to refer to Wales and the Welsh as Britain and British by the 12th century, they used the occasional reference to the Welsh as the ancient Britons. The Orderic Vitalis, for example, contains the note that the Britons were now called the Guali or Gualenses. References to the Welsh as Britons has been noted especially in contexts that aimed to convey their power and dignity, like in John of Salisbury's account of Henry II's defeat in North Wales:

Circumferat quis oculos mentis et intueatur quot et quales aduersarios ei Dominus suscitauerit ex quo aduersus Deum in depressione ecclesiae erexit calcaneum suum, et plane mirabitur et, si prudens est uenerabitur iudicium Dei qui non imperatores, non reges, non principes nationum ut ipsum domaret elegit, sed extremos hominum, Britones Niuicollinos, primo.

Let anyone turn his mind’s eye to view the number and the quality of the enemies which the Lord has raised against the king and, if he is wise, filled with reverence for God’s judgement: for he has chosen not emperors or kings or the princes of the nations to quell him, but chose first the remotest of men, the Welsh [sic] of Snowdon.

However, in other contexts where an English author wanted to target the 'immorality' or exasperating rebelliousness of the Welsh, they would more often use the term 'Welsh'. John of Salisbury again, writing to the Pope about the need to rein in the Welsh:

Et patentibus litteris uestris communiter ad episcopas Angliae, Waliae, Hiberniae, Scotiae destinatis praecipiatis, quatinus quod nos, urgente necessitate episcopi apud nos exultantis, canonice statuemus in Walenses, ipsi ratum habeant et sententiam nostram firmiter obseruent.

And by letters patent to charge the bishops of England, Wales, Ireland, and Scotland to regard as ratified whatever we, in view of the urgent need of the bishop now an exile in our house (Bishop Maurice of Bangor) may canonically decree against the Welsh, and firmly to obey our sentence.

Interestingly royal charters don't appear to make a distinction between Welsh and Cornish, which is probably fair as at the time they may have been as close to each other as some regional dialects of English. Here's a charter of Henry II in 1156/1157:

Episcopo Hereford’, abbatibus, baron(ibus), iustic(ie), uice(co)m(iti) et omnibus fidelibus de Herefordscire francis et angl(is) et walensibus salutem.

Greetings to the bishop of Hereford, abbots, barons, the justice, the sheriff and all sworn men of Herefordshire, French and English and Welsh.

And another from the same time period:

H(enricus) rex Angl(orum) et dux Norm(annorum) et Aquitan(orum) et comes Andeg(auorum) episcopo Exon’ et omnibus iustic(iis) et baronibus et uic(ecomitibus) et ministr(is) et fidelibus suis francis et anglis et wallensibus Cornubie et Deuonie.

Henry king of the English and duke of the Normans and Aquitanians and count of the Angevins to the bishop of Exeter and all his justices and barons and sheriffs and officials and sworn men French and English and Welsh of Cornwall and Devon.

So to summarise, the English considered the Welsh and other Brythonic-speakers to be of ancient British heritage but they seem to have considered themselves to have defeated the Britons and expelled them into the margins of the island. The idea of an indigenous claim to or future Welsh recapture of the whole island was merely an annoyance; comparing to the situation elsewhere, the Irish Book of Invasions (Lebor Gabála Érenn) does not really privilege the claim of earlier migrants to Ireland either.