It has been confirmed by Danny Sullivan of Google that the June 2019 core update has finished rolling out on June 8. The update began rolling out on June 3rd.

The core update took about five days to completely roll out. Updates like this take time since Google has a number of servers and data centers around the world. Due to this, it takes time for them all to get this update.

Here is a tweet by Barry Schwartz asking if the June core update is finished rolling out:

So the question is, how big was this update? There was some early data on it while it was still rolling out. There are some large publications including
the Daily Mail and CCN who is saying how bad the update hit them.

SourceBarry Schwartz