Others have mentioned the Baker-Feldman theorem and similar results coming from transcendence, which is the major source of weak effective general bounds in diophantine approximation. There is also the following paper, which deals with some special cases:
E. Bombieri, AJ van der Poorten, and JD Vaaler, Effective measures of irrationality for cubic extensions of number fields, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 23 (1996), 211-248.
I don't think it is possible to adequately survey a big area of research in a MO post. If we knew why it is difficult to make it effective we might be able to prove something.
I should also add that both the ABC conjecture and Vojta's conjecture (which is a generalization of ABC, I guess) imply effective (perhaps up to some constant) versions of Roth's theorem, so we kind of know what to expect.
BTW this question is a duplicate of Question related to Diophantine approximations and Roth's theorem
