I'm pretty sure neither musical was ever aired on TV let alone Live from Lincoln Center.
There was, apparently a concert of Tale of Two Cities that was filmed and aired on TV, but while James Barbour (the star of the show on Broadway) repeated his role, it is not the original cast. Michael York was in it as a Narrator, possibly because Barbour would have been close with him from when they did Camelot a bit before TOTC.
It also has a lead actress who looks like Rachel York (also from their Camelot) but seems to be Brandi Burkhardt (who i am unfamiliar with), and also has Natalie Toro who was one of the stars of the show on Broadway. The trailer calls it "direct from Broadway" so perhaps it was filmed soon after, as a way to get the show sold to regional and international producers, and maybe it is the full original cast?
I didn't know it existed, I kind of want to watch it now. Imdb says it's a "TV special", but I don't know where or when it aired. The Amazon page for buying the DVD says it aired on PBS, and is "featuring members of the acclaimed Original Broadway cast".
The is also a 47 min documentary "The Making of The Pirate Queen" that you can find on youtube, clearly made for TV -- I'm not sure where it aired, or even if.