Haven’t seen a thread about it posted here, but it has been confirmed that Overwatch 1 will be shutting down on October 2 - about 27 hours before Overwatch 2 goes live.
The following text can also be currently found within the Overwatch 1 client:
We’ve taken the Overwatch servers offline temporarily as we prepare to launch Overwatch 2. Servers should begin to come back online on October 4, 2022, 11 AM PT. Can’t wait to get started? Get equipped with the Watchpoint Pack, which includes the Season 1 Premium Battle Pass, two Legendary Space Raider skins, and 2000 Overwatch Coins—enough to pick up the Premium Battle Pass for Seasons 2 and 3 when they are released! It’s almost time, heroes… we’ll see you soon in Overwatch 2!
Presumably this will be displayed if you try and sign in during the 27 hours downtime.
EDIT: Changed the title to October 3 as its more correct than the news articles. Servers will shut down at about 8am PT on the 3rd. The 2nd will be the last full day of OW1.