They have never been in the same continuity.
Most people assume that they share the same continuity, and particularly that the Doom Patrol episode of the first season of Titans is a backdoor pilot. That's understandable since the show and the episode have mostly the same cast and, according to the Wikipedia article on the show,
On May 14, 2018, it was announced that the video on demand service DC Universe had given a spin-off of Titans, featuring the Doom Patrol characters
However, this dccomics.com article explains that
Showrunners Jeremy Carver and Sarah Schechter reveal that the continuity of the DC Universe streaming shows is a little more complicated than you think. (...) "They're in the same world," [Jeremy Carver] confirms, but then clarifies. "We're asking fans to come to this version of the show knowing that it is the same world, but it's a different continuity. While there are elements of Titans and the show’s introduction of our team, not all characterizations are the same and it's best just to come into it with a fresh look."
That choice is a way to simplify the production of these show:
"It’s so hard," Schechter says of making any crossovers happen, because while Doom Patrol films in Atlanta, "Titans shoots in Toronto."
"We can do the crossovers on The CW because they’re all in Vancouver and that makes it easier, but it’s still nearly impossible," she adds. "This would be really difficult."
She then explains that Geoff Johns "in particular has such an affinity for the Doom Patrol," and she thought what Titans did was "a really fun episode."
"But we were like, what would it be like if they had their own show?" Schechter continues. "When you start building out that whole world, you have to start from scratch. And also, it's a different show. Jeremy had to make it his own thing. You can tell that this show is so out of the particular alchemy of his brain. If he was constrained by the Doom Patrol as a side trip on Titans, it would limit him and we have to take those handcuffs off of him to feel free to create."
(Source: same dccomics.com article)
Note: there was at least one continuity issue between the series Doom Patrol and the episode of Titans. In the episode of Titans, Niles Caulder is seen walking normally and his back is broken by Raven using her powers, while he had been in the wheelchair for some time in the series.