Fiyero is a Time Lord.Because, let's face it, every section needs one. After the musical, when they're gone from Oz, the two run into the Doctor somehow. (Not sure which one, but it's likely Ten.) Elphie eventually gets killed. Fiyero, still a bag of straw, sets himself on fire; silly him, he did it in the TARDIS, and so he regenerates into someone who looks like Prince Edward. The Doctor kicks him out in Andalasia because he was getting way too angsty, and Fiyero takes Edward's place.
The Wizard is a Time Lord.It's never explicitly stated in the show that he's from Earth. The closest we get is him describing himself as a "corn-fed hick", however it could be that he simply hails from the countryside of Gallifrey.
Enjolras is a Time Lord.His other incarnations include AbbieHoffmann and RENT's Mark Cohen.
The Leading Player is a Renegade Time Lord/LadyHis initial incarnation resembled Ben Vereen in the 1970s, but he has been able to regenerate countless times since, and has even become adept at changing his sex. The other Players are his Companions and the stage is his TARDIS, which explains how some of Pippin is set in Medieval France. He has no name and is referred to simply as "THE Leading Player" the same way the other Rogue Time Lords are. He displays Time Lords' typical lack of fashion sense with his usual outlandish outfits, and likes to get his dirty work done, but in true Time Lord fashion, doesn't like to get involved himself, but gets his Companions to do it while he observes. Upon staring into the Untempered Schism he was one of the Ones Who Went Mad and decided he was going to kill some humans in the most fun and amusing way possible.
Dolly is a Time LadyHer business cards are psychic paper; they say exactly what she wants the person she's talking to to read.