Though you can probably get the reference more precisely correct (or perhaps having little feeling for German makes it harder to get the spelling right...):
[Blumenthal~1904] O. Blumenthal, {\it \"Uber Modulfunktionen von
mehreren Ver\"anderlichen}, Math. Ann. Bd. {\bf 56} (1903), 509-548;
{\bf 58} (1904), 497-527.
I'd wager that there is no literal translation of that work into English, in part for the well-known reason that throughout most of the 20th century professional mathematicians needed to be able to read (mathematical) German and French, with English as a third. Russian, perhaps, but there were political overtones.
As to the mathematical content: I'd expect (without looking again) that the early Hilbert-Blumenthal modular form work was quite naive in terms of several complex variables, and in terms of algebraic geometry (since theirs would have been exactly analytical geometry).
Siegel's "Advanced Analytic Number Theory" (in English, and available from TATA's website) probably reflects similar sensibilities to the early Hilbert modular forms stuff, as Siegel was known to not be "a fan" of "fancy math".
Also, Hecke's work, earlier than Siegel's, would have reflected a comparable worldview (but/and surely was/is in German).
My own old book from 1990 on Holomorphic Hilbert Modular Forms has a first chapter that attempts to recap a pre-Iwasawa-Tate-Gelfand-Jacquet-Langlands technical viewpoint, before upgrading to an adelic viewpoint. That is, that first chapter does not use any genuine algebraic geometry, no toric varieties, nor representation theory, nor ... almost anything. That is, it attempts to do Hilbert-Blumenthal automorphic forms in parallel to the 19th-century treatment of elliptic modular forms, perhaps to a considerably degree channeling those people.
An interesting point, already recognized many decades ago, is that Riemann-Roch ceases to be relevant. Nevertheless, there are immediate, special ideas applicable to arithmetic quotients that have no counterpart for general algebraic varieties, that allow proof of things comparable to what appeared to be consequences of Riemann-Roch. It is my impression (though I've not checked recently) that many of these very-specific tricks were known early in the 20th century. Perhaps not to Blumenthal, but to Hecke and Siegel at latest.
The most direct answer for you might be to look at Siegel's "Advanced Analytic Number Theory", which is in English, and though written several decades later than Blumenthal (and a few later than Hecke), probably is not tooooo far from their viewpoint, due to Siegel's conceptual conservatism.
