Timm couldn't use Hugo Strange either, because The Batman was using him. Even though Strange showed up several times on JL (non-speaking mind you) and appeared back on BTAS (where he uncovered Bat's identity). Timm reworked JL to have Batman match wits with Waller, instead of Strange.I really don't think people would have been confused by Nightwing showing up in Justice League while a totally different Robin appeared on TT, but corporations always assume viewers are stupid (and I think even kids wouldn't have been confused by stuff like Nightwing/Robin). But, from what I remember Timm also wasn't allowed to use Ted Kord/Blue Beetle, for reasons that weren't explained.
Anyway, hopefully embargo's won't be as big a thing anymore, since the general audience for superhero things might actually get treated as smart enough to understand that there can be several versions of the same character at once.
On TT, despite the show being mainly adapted from Wolfman and Perez's New Teen Titans; Kid Flash and Wonder Girl never make appearances alongside the main 5 Titans. Kid Flash shows up in one episode, but the other Titans are MIA. The reason being JL was using Wally as their Flash, so TT couldn't use Wally for Kid Flash. Even though Rosenbaum (Lex Luthor on Smallville) was voicing him on both TT and JL. Donna's rights were tied up with Diana's. Allegedly it took some doing to get WW on the JL show. On Batman Beyond, Timm used Big Barda, because he couldn't get permission to use Diana. Also, Batman never makes an appearance despite the constant reference to him by Robin throughout the show.
It's like execs think the audience is stupid and can't discern these are different interpretations. Just like you said.