how soon after a movie is released can the director release the "director's" cut or uncut version?

It varies. 

A director's cut may come out months after the theatrical release (like Batman V Superman Ultimate Edition) or years later for an anniversary release (like Blade Runner Final Cut).

It all depends on the studio.

Depends on many things. Mainly, if the studio thinks there's a demand/profit to be made from it. Usually for many movies they don't, and if the movie is part of a shared universe they may not want to release it on the grounds that their version of the universe is different then the director's. The Snyder Cut is the first time something like this has happened, and the main reason it did was because WB has basically given up on that version of the universe and they're going to be rebooting it, and they knew they would get a lot of attention/subscribers from releasing it because it had been a trending topic for a long time online. But anyway, sometimes a new cut can come out only a year later, like with Ghostbusters and Batman Vs Superman, and sometimes it can take many years, a decade or more. 

Educated guess? When the rights to the unedited rushes become the property of the director.