The long time story of the Anglo-Saxon peoples are that they hailed from the angles and Saxon tribes of modern day northern Germany. However, I do know that this is being debated and the extent of the migration being questioned. Would Alfred the Great's ancestry have been entirely Germanic or would he also have some Celtic origins?
I did my MA on the Adventus Saxonum (Arrival of the Saxons to Britain), so I'll have a stab at this.
You are right that the migration is being questioned. Initially we have the theory of mass migration, positing that a migration of incredible proportions took place, migrating women, children and men alike, to Britain. This is seemingly supported by the Romano-Briton Gildas in his De Excidio Britanniae, and then furthered by the Anglo-Saxon monk Bede in the Historia ecclesiastica gentis Anglorum. These two sources differ slightly, but they agree on some main points. These are:
The Saxons are a divine punishment for the Britons' sins. Bede elaborates on this point by stating that the British sin was that the Britons' didn't profess the faith to the Anglo-Saxons.
The Saxons drive out the British
The Saxons mass migrate in droves to Britain. Bede even states the capital of the Angles, Angulus, 'remained deserted from that day to this'. Though it's clear that by Bede's time, the Adventus Saxonum has become tantamount to a legend, and very few actual sources remain.
Both of these sources are unreliable, but they're the best we have. Gildas, whom we don't know much about, is writing sometime in the 6th century, this brings him firmly in line with the Adventus Saxonum dates, and even, if some scholars are right in saying he wrote De Excidio at a later date, would actually bring him in line with the formation of the famous Anglo-Saxon kingdoms. This makes it all the more strange that Gildas mentions that all external threats have been dealt with. Which seems very odd, considering he could well be writing a time when Anglo-Saxon kingdoms are popping up and making themselves stable.
Older historians chose to take Gildas and Bede at their word, mostly for lack of any other evidence, and at times seemingly supportive evidence (the destruction of Latin as a language on the island, and disappearance of Christianity). Gildas is very close to the source, but he may be talking about localised events, and may not necessarily know about the Britons' fate and treatment outside of that. As I say, we're not even sure when he's writing, let alone where. The most we get from him is '44 years after the battle of Mons Badonicus', which we also cannot date. Then for Bede, he's clearly writing far after the event (8th century), and it's clear he copies Gildas' version very heavily, whilst also tilting it just enough to showcase the Saxons as the heroes. This doesn't really help us much. Bede's piece is far more interesting and informative in showcasing the Anglo-Saxon identity and their collective memory of their own past, rather than how it actually happened.
Historiography has then shifted towards 'elite replacement' in the modern era, with two separate theories within this. We have discontinuity theory, and continuity theory. Discontinuity theory states that the Anglo-Saxons brought rapid cultural change through three main possibilities:
The British population was overwhelmingly killed, enslaved, or fled abroad (Brittany/Wales), as is proposed by Gildas and Bede
The British population instead remained, yet had their culture oppressed heavily, with Anglo-Saxon culture imposed on them, possibly via apartheid (apartheid is a good way for the Saxons to have amplified their genetics despite a small initial number of settlers)
The British population could have been 'Saxonised' through intermarriage or a self-conscious decision to switch to the politically dominant culture and language
That last point is very salient, as the foundations of the kingdom of Mercia is often debated. The term Mercia just means border people, and isn't a denotion of a certain ethnicity, such as Wessex (West Saxons), and was not heavily associated with any like Northumbria (Angles). Given its position, it could well have been a Britonnic kingdom turned Anglo-Saxon via its neighbours influence/Royal marriage. This is an interesting view considering many of Mercia's Royal lineage has 'celtic' names within it, especially nearer the beginning, such as Cnebba, Icel, Creoda and possibly Penda/Peada. This would serve to show that the Celts were not wiped out, at least in Mercia, and that a strong Celtic influence remained, although muted and suppressed by the stronger Saxon culture. This is called 'cultural invisibility' where a culture's impact on history is invisible due to a lack of written and material evidence. Germanic material culture rapidly replaced Roman material culture after the collapse of the Western Roman Empire, and the Saxons were fantastic at oppressing the Celts when it came to writing.
Continuity theory, however, states three other options:
The Anglo-Saxons were largely no different from prior rulers in Britain, and thus did not bring about any rapid cultural change for the average Briton
The Britons may have kept their culture for a very long time and did not necessarily 'Saxonise' in any meaningful way for centuries after the initial migrations
The Britonnic population could have remained largely undisturbed by the Anglo-Saxon arrivals, who left them alone, and allowed them to live as they had done prior to it.
There is, of course, a third model which is a mixed based model which states that any/all of these could be true for various different parts of Britain.
And that is probably most likely what happened in our case here. As we saw earlier, Mercia may well have had Celtic roots, but is that the same for Wessex?
Well, the original founder of Wessex, Cerdic, is a possible Celtic name, as well as his successors Cynric and Ceawlin. These were rulers of the so called Gewissae, which is an Anglo-Saxon origin name, which gradually turned into 'King of the Saxons' by the time of Caedwalla in the 680s. This royal house then became the beginnings of the kingdom of Wessex. So, much like Mercia, we may be seeing a hodge podge of Celtic and Saxon influence.
Now, we don't know necessarily what Alfred's lineage is. He does tell us, but it's whether we believe him. The Anglo-Saxon Chronicle MS A, or the Winchester Chronicle, is the earliest version of the Anglo-Saxon Chronicle, and was most likely made in Wessex under the reign of Alfred. It's prefaced with a genealogy of Alfred going from Cerdic all the way up to Alfred, which contains various interesting figures. It contains a euhemerism in that it states he's a descendent of the Germanic God Woden. The ties to Cerdic are very important here, as almost every Kings reign is suffixed with 'whose ancestry goes back to Cerdic'. This attempt to legitimise and bolster Alfred's lineage is a very conscious one. His grandfather, Ecgberht, was a very powerful Anglo-Saxon king, and is the first real claim to the Wessex throne that Alfred has. Before this, Ecgberht's line last has a royal member in Ingild (672-718), and no actual Wessexian King for eight generations. Although Ealhmund, Ecgberht's father, had been king of Kent in 784. Thus Alfred linking his lineage back to Cerdic is a conscious political decision to legitimise his house. Alfred himself is only on the throne because his brothers son was too young to rule in the face of the Viking threat, and this is of great concern for Alfred. His brothers son actually does try to stage a rebellion against Alfred using Viking help to claim the throne much later on, but fails miserably.
If this genealogy is true (and it's highly doubtful), Alfred's lineage goes back to Cerdic, who is most probably of Celtic descent. Meaning that, yes, Alfred does have some Celt in him. The real answer though is, we don't know. Alfred is linked to all sorts of figures in the Anglo-Saxon Chronicle, and in the year 855, when explaining Aethelwulf's background (his father), Alfred is linked to legendary Kings Geat of the Geats of Beowulf fame, and Sceldwa and Heremod, two legendary Kings of the Danes. Almost all entirely fabrication, but is an interesting way of Alfred perhaps putting himself forth as a possible legitimate leader of the new Nordic people in his lands. The genealogies of Alfred also go back to God, which whilst irrelevant here, is interesting considering many historians now believe Alfred viewed himself, or rather wanted others to view him, as a new 'David' from the Old Testament. This isn't necessarily ground breaking news; many kings linked themselves to David, but Alfred had a smoking gun in the Vikings. Many of the tall and dark tales of the Old Testament would have seemed very apt for the average Saxon to help explain and understand his current horrible predicament, and Alfred comes out of this with a very new dynasty (like David), whilst seemingly being supported by God himself with major victories (again, like David). Alfred would then use these analogies to push for a new state, one of a united Anglo Saxon nation (the first real sightings of a state of England). So these genealogies were very important politically. There's a lot going on in them, and we shouldn't take them at face value. They're all telling us an intricate political story
Though what's most important, is that despite what his dna would make up, he was culturally Saxon. We cannot tell in the graves of West Heslerton, long since thought an 'Anglo-Saxon site' just how those people identified. They, due to cultural invisibility I mentioned earlier, could have identified as Celtic, Roman, or even Norse, but we have no way of knowing unless we revive them and ask. Alfred however clearly feels Saxon, and starts the ball rolling for a united 'English' nation. Though this 'English' nation almost certainly has more Celtic roots than it realised
I shall note my sources in a bit, just struggling as I've written this on a phone and with this knowledge a little bit rusty in my mind. Please feel free to ask any questions, and I hope this suffices