The Exorcist: Believer is in the same continuity as the other Exorcist movies beyond the original, stated by the director. This made me interested in learning more about the timeline.
The problem comes with the prequels. They're different films telling the exact same story in a different way. It's confusing, but basically the films feature the same characters in different ways with entirely different scenes. There's also some issues between the prequels and the first two films. I'm planning on watching all the films as if they were the same continuity though, just to see if it could fit in any way. The prequels are likely going to be different continuities to one another, so I guess one of them is supposed to be thrown out.
At the very least, the first three films are confirmed canon. If the new film says the first movie is set in 1973, that's a small retcon to the third, but either way, it's canon.
There's a crossover character with The Ninth Configuration apparently, and it's apparently explicitly related to The Exorcist in general.
The TV Series is a sequel to the original film, but a character dies in it that is alive in the new installment so it can't be the same continuity as the films. So, that's absolutely cut.
Film Timeline
1947 - Dominion: Prequel to the Exorcist [Schrödinger's Canon]
1949 - Exorcist: The Beginning [Schrödinger's Canon]
1975 - The Exorcist
197? - The Ninth Configuration
1979 - Exorcist II: The Heretic
1990 - The Exorcist III
2023 - The Exorcist: Believer
TV Series Timeline
1975 - The Exorcist
2016 - The Exorcist 1x01-1x10
2017 - The Exorcist 2x01-2x10