Not exactly, but kinda.

This is a slightly more complex question than you might realise.

The issue here is that languages, all languages, are constantly in a state of change. The English you speak is a tiny bit different from what your folks speak, even if you think it's exactly the same. There are small but measurable differences. These add up over time to create much more obvious shifts.

However because it's happening in (usually) such tiny steps, you can't actually say "this is the point where it shifted".

What we can say, though, is when there were events that led to more significant shifts.

One of those major events in the case of English happened in 1066 with the Norman Conquest. That's the point that we usually stop calling the language Old English and instead call it Middle English. 1204 is another significant year, as that's the point that we can say that the already ongoing Englishification of the formerly-Norman-French spoken by the upper classes finally got the final nail in its coffin. That was the year that Normandy was conquered by Philip II. Anyone in England with ties to Normandy effectively had those ties cut in that year. The convergence between Norman French (Anglo-Norman) and Early Middle English can be marked off as ending there (well, within a generation of there).

All that said, it's the period of Middle English where we as modern speakers can start to see the language as looking somewhat familiar. This is not usually the case with Old English. For that reason, we actually can give a century to answer your question. The 12th-13th century would be that.

But again you should remember, change is happening all the time, even now, and rarely does it happen in a way that we'd call a shift, at least if there's any intent implied behind that term. Yes, there are language policy issues and governmental efforts to push major shifts, and there are plenty of other factors that can make the change happen much more abruptly. The 1066 answer isn't actually a clean point but rather a convenient reference point, while the actual changed that caused Old English to become Middle English took generations. 1066 is just the thing that motivated the specific direction they went.

tl;dr: 1066 (kinda)