Bruce Timm seems to be the main connecting factor behind the DCAU Justice League cartoons and Batman Beyond.
Yes, he and Paul Dini are largely credited with Batman: TAS, Superman: TAS, Batman Beyond, and the Justice League TV Series and their shared universe.
He's also one of the producers behind The Dark Knight Returns film.
After Justice League Unlimited was cancelled, he moved to work on the DC Universe Animated Original Movies. He directed several of them, and has been a supervising producer on all of them. This changed earlier this year when he moved on to develop something else for Warner Bros.
Does this fit into the continuity of the two aforementioned cartoons? I realize it is based on the comic series of the same name, but it also seems as thought it should fit between the two.
The DC Universe Animated Original Movies are all meant to be one-off (excepting the 2-part TDKR) short films that adapt classic comic book stories. Each one is in its own continuity, as they don't do 100% faithful adaptations, so they're even not in the same continuity as the original source material. They're not meant to be continuations of the Timm/Dini cartoon universe or any existing canon; they could roughly be considered an animated version of Elseworlds.