Rights-wise, the new show is already paid for. And Constantine's creators won't see any royalties from it for a long, long time, if ever. Because it's part of the movie deal that gave us the Keanu Reeves film.
This option apparently rolled out of the already-paid-for option for the CONSTANTINE movie in the 1990s. Thus, we'll only see $$ waaaay down the road, it appears, IF this series makes it to being a series. If it makes money. If it trickles down.
Will update you if/as we learn more.
But we will see $$ from any comics/graphic novels sold from the spillover of interest, FYI.
Justice League Dark, the proposed Guillermo del Toro feature with Constantine as the main character, would also have featured characters like Swamp Thing and Deadman, thus requiring a new deal.