While any version of English will likely have considerable influence from French and/or Norse, it needs to be said that English borrowed many, MANY more words from a Gallo-Roman dialect than any other Germanic language (inclusive of Dutch, Luxembourgish, etc.). Furthermore, whilst French loanwords into such languages do exist, they entered the language due to proximity and prestige alone; the English language had its metaphorical arm twisted by Norman overlords into throwing away perfectly good word sets and SUBSTITUTING them with Norman French equivalents (hence the Germanic-Latin register difference).
There's no doubt that, barring a Norman victory, there would still be borrowed words from French into English, but it would be a much smaller, more evenly "spread" set of words as opposed to the overbearing number they contributed IOTL, and likely without that perceived "formality" that Anglo-Norman words hold due to a more diverse subject set.