The name Angle, in regards to inhabitants of Britain, first appears in Procopios' Wars (8,20) as describing the Angiloi living in the island of Brittia alongside two other peoples, namely Frissones and Brittones all described as insular natives.

Procopios doesn't seem to have a clear picture of what happens in Britain in the VIth century, even having trouble separating informations about the island of Britain (Great-Britain) and the western part of Armorica which was beginning to be called Britain as well (Brittany), mentioning the latter as Arborychoi (a likely deformation of Armorici) without identifying them as Bretons. The reason might be he was depending either on ancient texts about the regions (hence mentioning the Hadrian's Wall and the mirabilia attributed to the northern parts of the island commons in mentions of Britain in Classical and Late Antiquity) and especially from informations given by a Frankish embassy in Constantinople boasting of the Merovingian overlordship over northern-western Europe, particularly over Brittia and Brittania in addition of Gaul and Germany.

Those claims can appear to be quite boastful (and they certainly are, as Procopios asserts) but aren't entirely fanciful either, several elements pointing at these regions being part of the Frankish periphery and having a relationship with Merovingian kings comparable to which they had on Germany (that is a more or less direct overlordship over peripheral polities trough military means, gift-diplomacy, local support, etc. extracting tribute, military service, control of trade, etc.) not only in Brittany but also possibly on southern Britain. (The when and the how of it being a whole different matter I don't want to derail this question with, better discussed in another post)

It's also interesting to notice that the names "Angles" itself had virtually fell into disuse in Roman sources since the IInd century, last mentioned by Ptolemy (although this could actually be a mistranslitteration of a different group). During Late Antiquity and the long list of Barbarian peoples or ensembles, they're simply not appearing as such, quite possibly integrated within broader North Sea groups : it doesn't mean Angles simply disappeared as a people, but that they might likely have been integrated within one of the Barbarian entities forming alongside the limes by the IIIrd century (e.g. Sigambri or Tencteri as Franks).

Still, the question remains why Procopios would have resorted to an antiquated term : it's not impossible that an Anglian identity might have been preserved unmentioned and that Procopios' text is reflecting that, but we also can point to the authors of the Wars having only second-hand informations about the region he have trouble having a good picture of.

Then we have to get a look at the possible main informers, Merovingians, and be sorely disappointed, because contemporary Frankish sources do not mention Angles : what they seem to do, however, is using antiquated terms taken from Tacitus, Pliny or Ptolemy as Frisii, Varni or Eudoses (possibly Eucii and Euthiones mentioned by Theudebert and Venantius Fortunatus as under Merovingian overlordship) similarly re-emerging over centuries.

There again, it does absolutely not disprove that Germanic populations settling Britain claimed an Anglian identity : but it does begs the question whether or how much Franks purposefully made use of an antiquated term to legitimize their power (or claims of) over peripheral peoples to both Gallic elites and the court of Constantinople, trough a classical and prestigious Roman framework and especially the ethnographic best-seller that was Tacitus' De Germania.

This would be particularly interesting as non-Frankish contemporary sources (Gildas, Zosimus, Constantius, the Gallic Chronicle, etc.) almost systematically goes with "Saxons" to name raiders and newcomers in Britain (as well as the related peoples settled in Picardy and Normandy), one that does have a continuous use from Late Roman sources, pointing that contemporary peoples had the notion of a distinct set of identities from Roman or British in a different scope Procopios and his Frankish sources would have.

The wave of Early English settlement in the British Isles in that fuzzy period between Late Antiquity and Early Medieval was characterised by a diverse range of peoples, but by the 7th Century, the writer and historian Bede, in his Historia Ecclesiastica, had narrowed it down to three main groups: the Saxons, the Angles and the Jutes. Bede delineates distinct areas where each people supposedly settled and ruled and, regardless of what the situation actually looked like on the ground, other historians essentially took him at his word for the next 1200 years.

Somewhat to Bede's credit, the rulers of Wessex - the Kingdom of the West Saxons - do tend to identify themselves as Rex Saxonum, but the ethnic identifiers are far less straightforward in other areas. In Kent, for example, which Bede says was a Jutish kingdom, the kings from our earliest written record identity themselves as Cynges Cantwaras and never explicitly as Jutish. In similar vein, kings of Mercia (said by Bede to be Angles) typically identity themselves in charters as Rex Merciorum rather than Rex Anglorum. Our surviving charter evidence for another "Anglian" kingdom, Northumbria, is incredibly sparse, but our one surviving charter again identifies a Rex Northumhymbrorum rather than a Rex Anglorum.

Nonetheless, Bede's idea of a Anglalond with its Angelcynn or gens Anglorum seems to have stuck around, at least for the Saxons. When Alfred of Wessex and later his son Edward expand into previously-independent kingdoms in the couse of "liberating" the Danelaw, they begin to style themselves as Rex Anglorum et Saxonum to illustrate their control and unification of both their original Saxon kingdom and the supposedly "Anglian" areas with which they had allied and unfied (such as Mercia) or conquered and liberated (such as East Anglia) and the term quickly becomes abbreviated to Rex Angolsaxonum. The term itself doesn't last long; it's used in a handful of charters before being replaced by the far easier and more comfortable rex Anglorum, but to 18th Century historians, "Anglo-Saxons" was a more useful term.