There are applications of Thue--Siegel--Roth and related results to this question (but it seems that they use in some form or another the rational approximations already present in Mahler's work).

For example, Baker in "On Mahler's classification of transcendental numbers." Acta Math. 111 1964 97–120,
shows that this constant is not a U-number using a result he describes as "This extends a theorem of LeVeque [..] which itself is a generalisation of Roth's Theorem"

However, then for the proof of this application it reads "It is clear from the proof of these results that the hypotheses of Theorem 1 are satisfied..." Where 'the proof of these results' refer to Mahler's.

And much more recently, Adamczewski and Bugeaud in "Mesures de transcendance et aspects quantitatifs de la méthode de Thue-Siegel-Roth-Schmidt." Proc. Lond. Math. Soc. 101 (2010), no. 1, 1–26 generalise this result of Baker together with a classical result of Ridout to then show that the generalizations of the Champernonwne Constant (any base, any suitable polynomial) are all S or T numbers.
See section 3 of the paper, in particular Théorème 3.2. But again the proof uses the rational approximations of the original transcendence proofs (by Mahler); cf. the final paragraph of Section 3.