Alternative England
Some interesting comments here but I'm not sure I agree with many of them. There seems a basic assumption that Harold Godwinson would, had he survived Hastings, have been a "strong" king. I'm not sure of the evidence for this and it conveniently neglects the nature of what happened in the years before 1066.
During the 1040s and 1050s the Godwins effectively usurped the Saxon throne from the descendants of Alfred and Aethelred. Any independent assessment of the rival claims to the English throne in January 1066 would not have had Harold Godwinson's as the best claim. Indeed, it's fair to say William of Normandy had the better claim though the strongest claim was probably that of Edgar (known as the Aetheling).
The problem for Edgar (and William) was that the Godwins had manoeuvred their supporters into the Witan ensuring that even though a child from the marriage of Edward and Harold's sister had not been forthcoming, the Godwins would still control the destiny of the throne. Thus Harold's accession after Edward's death was assured.
The other thing that is often forgotten is how wealthy mid-11th Century England was in the context of western and northern Europe. With a generation of peace, England had prospered via English silver (used to pay off the Danes) and the wool trade with Flanders. Had England been the economic backwater so often portrayed, why would BOTH William and Harald Hardrada choose to invade ? The possibility of acquiring via conquest a wealthy country like England would have been irresistible and so it proved.
For Harold Godwinson, a victorious 1066 would have strengthened his position but not secured it. First, Edgar was still alive and in league with the Scottish king. It's highly likely that Edgar would have tried to take the English throne with Scottish help. Tostig (presumably dead at Stamford Bridge) was linked to the Danish royal house while William's (presumably) widow, Matilda, was linked to Flanders. Far more serious, potentially, than any of these threats was the possibility of revolt from within the extended and numerous Godwin clan itself. Men like the Earls Edwin and Morcar could have allied with Edgar. It was not unknown for the Godwin family to turn on each other and while Harold might have kept order for a while, there are no guarantees this would have lasted especially as he grew older and weaker.
One other thought. Given the mood of the time, I suspect that Harold's view after a victorious Hastings would have been that having failed to take England, William had forfeited Normandy to him. With William dead, could his young sons Robert and William Rufus hold the Duchy together ? The history of Normandy pre-William suggests otherwise. Moreover, could Harold, with the assistance of English money and favour, have undermined and usurped the Normans giving us, if you will, a Conquest in reverse with English forces landing in Normandy and with the assistance of men like Count Eustace of Boulogne or in league with the French king, overthrow Normandy ?
The argument that a Saxon England would somehow abstain from Continental Europe is, in my view, ridiculous. England was economically linked to Europe and, as I've argued, Harold might have seen a rold for England in Continental Europe. I could imagine a later conflict with Flanders for example.
To pick up some points made earlier, we could see Harold in league with the Emperor against the Pope in the mid-1070s.
All of this pre-supposes a vibrant and successful Saxon kingdom under Harold. It's also conceivable that, wracked by internal division, the fall of the Godwins would drag England into a protracted period of anarchy in the 1070s and 1080s. Indeed, we could imagine a smaller England emerging with the Scots and Welsh having gained territory at England's expense.
