Blizzard Entertainment has said that October 2 will likely be the final day that fans can jump into Overwatch. As stated to press members in a recent briefing call, we have been told that the Californian developer is planning on transitioning between Overwatch and Overwatch 2 through a massive 27-hour downtime period that will mean that the original game will no longer be playable over a day before the launch of the sequel.
It should be said that this is working on a Pacific time zone, meaning the exact dates and times might differ depending on where you are in the world, but this won't change the fact that Overwatch will be going down around 27 hours prior to the release of the sequel.
In terms of what players can do during this downtime, there will be pre-download options for Overwatch players as they transition to the sequel, and likewise there will be a variety of checks to complete heading up to this downtime period, which Blizzard will communicate with players in the next couple of weeks.