Advances and difficulties in effective version of Thue-Roth-Siegel Theorem

The non-effectivity, as far as I understand, is already present in Thue's Theorem, thus to understand it, one can look a the proof of the latter. The issue is roughly that, to show that there are not many "close rational approximations" $p/q$, one starts with the assumption that there exists one very close one $p_0/q_0$, and show that this very good approximation "repulses" or excludes other similarly or better ones. This of course doesn't work if the first $p_0/q_0$ doesn't exist... but such an assumption also gives the result! The ineffectivity is that we have no way of knowing which of the two alternatives has led to the conclusion.

There is a well-known analogy with the Siegel (or Landau-Siegel) zero question in the theory of Dirichlet $L$-functions. Siegel -- and it is certainly not coincidental that this is the same Siegel as in Thue--Siegel--Roth, though Landau did also have crucial ideas in that case -- proved an upper bound for real-zeros of quadratic Dirichlet $L$-functions by (1) showing that if there is one such $L$-function with a zero very close to $1$, then this "repulses" the zeros of all other quadratic Dirichet $L$-functions (this phenomenon is fairly well-understood under the name of Deuring-Heilbronn phenomenon), thus obtaining the desired bound; (2) arguing that if the "bad" $L$-function of (1) did not exist, then one is done anyway.

Here the ineffectivity is clear as day: the "bad" character of (1) is almost certainly non-existent, because it would violate rather badly the Generalized Riemann Hypothesis. But as far as we know today, we have to take into account the possibility of the existence of these bad characters... a possibility which however does have positive consequences, like Siegel's Theorem...

(There's much more to this second story; an entertaining account appeared in an article in the Notices of the AMS one or two years ago, written by J. Friedlander and H. Iwaniec.)

This is a kind of a late response, but the OP said "That is, the result nor its (original) proof provides any insight as to how big the solutions (in q) can be, if any exists at all, or how many solutions there might be for a given α and ϵ," and no one has addressed his comment about how many solutions. In fact, Roth's proof combined with a simple gap principle (and a lot of bookkeeping) gives a completely explicit upper bound for the number of solutions as a function of ϵ and the height of α. One can find versions of this in various places, including my paper: A quantitative version of Siegel's theorem: Integral points on elliptic curves and Catalan curves J. Reine Angew. Math. 378 (1987), 60-100. I have a vague recollection that Davenport may have been the first to point this out (maybe just for Thue or Siegel's theorem). There are also deep generalizations giving upper bounds for the number of exceptional subspaces in Schmidt's Subspace Theorem, see for example: A quantitative version of the absolute subspace theorem, J.-H. Evertse and H. P. Schlickewei, J. Reine Angew. Math. 548 (2002), 21–127.

There has been progress towards effective Roth's theorem. Notably, Fel'dman was first to prove an effective power saving over Liouville's bound.

In the nutshell the source of ineffectivity comes from the following kind of argument. One obtains a sequence of positive real numbers $x_1,x_2,\dotsc,$ with the property that the product of any two distinct $x$'s is at most $1$. It immediately follows that the sequence is bounded, but of course this information does not yield any actual bound. In the Thue's proofs, one argues that no two rational approximation can be very good simultaneously, which is where such a sequence of $x$'s arises.

In my opinion, a good introduction to effective methods in transcendental number theory is in the notes by Waldschmidt.