Doing this a bit back to front...
1066-England, its borders roughly as they are now, is best described as Anglo-Danish rather than Anglo-Saxon. Harold Godwinson was Anglo-Danish (his mother or grandmother was of Danish stock).
Hastings is a very close fought battle (all day battel at a time when most only lasted 2 hours) and even allowing for the fact a good part of the English army was knackered, it would only have taken a bit more discipline on the right wing(which it would have had if Harold's brother, commanding the right wing had not been killed) and William would have lost the day or at least not won, which would have amounted tot he same thing-battered Norman army, in retreat, no reinforcements, chased back to its ships as more and more English reinforcements arrive...
Part of the reason for the success of the Great Army was there was a civil war raging in Northumberia. No civil war and there is a strong possibility the Great Army may not have been successful or not as successful as it was. Of course, the Great Army destroyed the Anglo-Saxon Kingdoms except Wessex, this in turn is considered to be what led to the creation of a single England (the theory is, very simply, without the Vikings no one of the kingdoms would have been powerful enough to overwhelm the others, think the politics of the three megapowers in '1984').
Northumberia v Picts. If the Picts had lost then it seems likely that Northumberia would have seized their better territory and left them with the more highland bits (of little value to Northumberia). In turn this would have left the Picts weaker and an earlier surrender/take over by the Scots, who might themselves now be limited to the Highlands, unable to break into the lowlands. The now much stronger Northumberia continues to dominate, perhaps it remains the dominant power and so Mercia never achieves greatness (I think largely gained at the expense of the weakened Northumberia). This might make some changes in the south-if Mercia is tied up holding off a powerful Northhumberia their allainces with the Welsh might not come about. Freed of the Mercian threat, it is possible one of the southern kingdoms such as Wessex, Sussex or Kent might come to dominate the others (historically the Mercians domianted them), so instead of the Kings of Kent and Sussex submitting to Merica, they might submit to Wessex instead. So we end up with an increasingly powerful Wessex in the south, Northumberia in the North (who has taken over the Strathclyders and other 'British' kingdoms in the north) and Merica and the East Angles cuaght in the middle. A stronger Wessex might overwhelm Dumnonia/Cornwall sooner, perhaps even before the Vikings turn up. The Vikings still end up taking over the north but facing a much stronger Northumberia are much weaker when they attack the now enlarged Wessex who defeats them but sooner. Danelaw is smaller and the conquest of it is completed by Edward (next king after Alfred) rather than Athlestan (Edward's son).