How much is known about the Weyl character formula for quantum groups? More specifically, has the formula been generalized to the general setting of deformed coordinate algebras $\mathbb{C}[G_q]$ of semisimple Lie groups and their associated flag varieties? I am most interested in the nonroot of unity case.

The question still feels a bit vague to me, but this started to get too long to be a comment. There are a number of issues: There's the question of the definition of $\mathbb C[G_q]$, and say an analogue of the PeterWeyl theorem, and there is also the issue of doing this at a root of unity case, or better studying things integrally. For this say, a recent paper of Lusztig gives a definition of a quantum coordinate ring for any (finite type) root datum, which specializes to the KostantChevalley form. AndersenPoloWen and others have studied studied quantum induction functors which correspond to taking global sections on the classical flag variety, and these might be what you want (RyomHansen also proved a version of Kempf vanishing in this context for example). This was also more recently taken up by Kumar and Littelmann in the context of studying Frobenius splitting. Finally there's the issue of understand quantum flag varieties as noncommutative spaces as in the previous comment, for which along with the LuntsRosenberg paper, there is also more recent work of Backelin and Kremnizer. 

