How might have the English language been different had king Harold's forces won the Battle of Hastings?

French and Latin had words and terms that Germanic/Danish languages didn’t have, so eventually Romance words would have been adopted by the English. But certainly we would be much closer to Germanic than to Norman.

We would be eating roast cow, not roast beef.

Well—that’s assuming that the Normans didn’t win another one insead.

But yes, there would be much less Latin influence evident in English, since most of that came by way of French. Our language of law, justice, government, military, and other major social institutions is heavily Latin-derivative because the Normans brought those institutions with them.

The really basic words, though—heart, hand, hearth, home, love, life, and other essential expressions—tend to be those with Germanic roots. The French words tend to be more formal and a little detached; for example, residence or domain rather than home.

I’m guessing that English, if it persisted at all, would just be a minor local dialect now and not the world’s dominant language, the language that people everywhere use to communicate with others who don’t share their mother tongue.