And if Britain is overrun by the Norse, and significant numbers of Anglo-Saxons cross the Channel in search of refuge, their numbers could help butterfly away "Normandy," meaning the Danes in Britain would be constantly subject to raids and the threat of invasion from the south. I could even see a Danish England as the catalyst ITTL for the First Crusade in the late 9th-early 10th c.
There's no realistic way to have Norse overruning Anglo-Saxon entierly, only because of numbers.
OTL, Danes managed to take York essentially because of the civil war in Northern England (admittedly, Anglo-Saxon England knew several war between their kingdoms) that weakened the country and because of the underpopulated region surrounding York.
And they managed to submit Mercia, a kingdom that was importantly beaten by Wessex decades ago.
At some point, they find themselves against a Wessex whom power had managed to raise thanks to Mercia (that became vassalized) and the support of other Anglo-Saxons.
Now let's imagine that a Dane Overking emerges and manage thanks to some PoD to use a sudden and unlikely opportunity to make his supremacy acknowledged (aka, not an unified kingdom by acknowledgment of Anglo-Saxon petty kingdoms of their vassality) : Anglo-Saxons are more likely to stand and would be likely autonomous in such scenario.
Even in the lands directly ruled by Norse, I don't see why a conqueror with few reinforcement and ruling in a country weaken by decades of war would want to weaken even more what was his own by removing Anglo-Saxons for every situation of power (well, unless he want to rule on seagulls and stones, that is).
In any case, allow me to introduce you the Late Carolingian world : It's indeed a troubled place, plagued with dynastic wars.
Several hit and run raids are led victoriously by Norse, Saracens, Magyars; some are defeated.
But the issue isn't Late Carolingians are unable to rise an army, or to resist raids. It's they don't have time to gather great armies and to move against raiders quickly enough critically if they weren't the only one trouble makers and that once Norse settled in safe places (Noirmoutiers Island, by exemple) they were really hard to expel and that even victories wouldn't prevent new raiders to appear.
Now, we have Anglo-Saxons that come in significant numbers to settle in the lake IX century. They won't certainly be considered as allies against Vikings, but another breed of invaders, just like them, Saracens, or Magyars.
Everything depend on where they try to land :
- Nesutrian coast : The region is plagued by Viking raids. Not the wisest choice.
- Brittany : The same, with not-that-welcoming Breton nobility.
- Frisia : Europeans discover shishkebab far ealier than OTL. This is simply no way Late Carolingians let a wandering people outside their control settle in one of the most strategic place of their kingdoms.
The best outcome would be a "Hispanii-like" feature, where Anglo-Saxons submit to local nobility (such as March of Neustria) and are quickly merged with the surrounding population, effectively having no real effect.
See, Normandy wasn't a military conquest strictly speaking : while coastal and valley parts were more or less regularly settled by Norse already as winter bases (allowing them to have contact and beginning already to merge with local population) Rollo being count of Rouen in 911 is a result of the "frank-isation" that concerned all settling Norse in Francia since the IX century (as Harald Klak conversion) where finally the Scandinavian features were rare in continent compared to what existed in England.
The point is : with or without Anglo-Saxon coming in Francia, *Normandy is likely to be created, at least as keeping the "Norman" March of Neustria that pre-existed to OTL Normandy.
So, basically :
- No way that Danes can overrun all of Anglo-Saxon England. At the very best, they can vassalize lesser Anglo-Saxon kingdoms as Western Mercia instead of Wessex.
- Danes wouldn't chase a significant number of Anglo-Saxon out of their lands
- Eventual refugees would only come in small number and immediately submitting to Carolingians and/or local nobility that would relocate them. They end quickly by merge with local population.
Would have they fled more realistically, they would have gone as OTL after the Norman Invasion in Ireland, Scotland, maybe Wales.
- Normandy still happen, one way or another.
Finally, for the Crusade part : OTL the process that lead to the appearence of Crusade was due to several conjugating factors.
- Militarization of nobility and warlike way of life
- Popular and clerical reaction against feudal wars
- Subsequent efforts to conciliate noble way of life and religious reforms to enforce peace, by making violence legit if it was for protecting Christians
- Great troubles in Middle-East due to Turkish invasions that challenged the pilgrimages (less due to persecution than anarchy) and call from ERE to help.
Having a First Crusade directed against England would need for persecution of Christians (when Danes seems to have collaborated with local high clergy quite well), to butterfly the most important pilgrimage of western Christianity in Jerusalem or to butterfly Turkish invasions.
Not talking having a papacy strong enough to enforce *Gregorian Reform or call a Crusade when with this PoD, it could be relatively easy to delay (or even butterfly it)