Well, this is an interesting question, I must say!
As has already been pointed out earlier, this involves many centuries of divergence-nearly a millenium in this day and age-from our own world so there are many possibilities.
That said, though, there are a few things I'd like to explore briefly.
Firstly, indeed, what of an English language significantly-but not dominantly-influenced by Norman French? As I recall, just under a century and a half ago, such an idea was actually postulated by the celebrated French scholar Jean-Pierre de Hautemont; his speculative work, I may add, was inspired in no small manned by the unique dialect mainly spoken by residents over here in the eastern central areas of North Veridia-namely what's now the modern nation of the Veridian Commonwealth(yours truly is a proud native of New Perth), and some of the small nation-states adjacent to it, such as Pouhatan, Cushata, and Mescoka.....though it really only formally emerged after the fusion of many of the former English and French colonies here during the early 18th century, and Norman French wasn't the only French dialect that had significant representation here, so 11th century England, needless to say, would see something slightly different.
Secondly, a Norman victory, I believe, would likely prevent both the creation of Norwall and Sowall (that truly was, as I've discovered, one of the major reasons why the Welsh had such poor relations with England for the next 250 years after they regained their independence in the mid 13th Century, and even then, some significant cultural distrust and prejudice lingered for another 4 whole centuries afterwards), as well as the 167 year long Great Unification between England, Denmark, and certain of the German states, especially as none of the major French kingdoms had particularly good relations with even Denmark, let alone England or most of the German states-save for Aquitaine & Bavaria for a time-and wouldn't until near the end of the 15th Century in our world. If anything, I'd suspect that this alternate England would be more likely to cultivate better relations with some of the French nations, simply because the Normans-although their ruling class was largely of Scandinavian extraction-were still a decent bit closer to their fellow Frenchmen than any of their neighbors even in 1066, at least as far as I'm aware.