Source: Blizzard
So, it looks like those who don't upgrade to Overwatch 2 won't be able to play with those who decide to play the new game. In fact, the original title will be replaced by Overwatch 2, rendering it unplayable. This update comes from a Reddit AMA with Overwatch 2's development team held on June 22. One of the questions asked was whether the new game would "replace" Overwatch's PvP or shut the game down entirely.
"We're using the term 'Early Access' to indicate this is just the start of many new things coming to the game," game director Aaron Keller said. "We're launching with new heroes, maps, and features, but there are even more of these coming seasonally—every nine weeks. We recently released a roadmap detailing some of this content, with a new hero coming in Seasons 1 and 2, and a new map in Season 2. Additionally, larger pieces of the game that have always been a part of the vision for OW2 will be released to the game as part of the live service, including the launch of the PvE campaign next year. When OW2 launches on October 4, it will be a replacement for the current Live Service."
This response suggests that Overwatch 2 will replace the original, which contradicts what former game director Jeff Kaplan said about these two working in a "shared multiplayer environment." One good thing about this sequel is it will be free-to-play, and you will carry your progress over to it. Overwatch's 6v6 setup will be retired in favour of the new 5v5 matches format.