In the case of die Übersetzungen, one could argue that it is an implicit aforementioning (after Duden-Grammatik, § 386). There is an "entirety-portion relation" (ibid.) between Banca d'Italia from the preceding sentence and the Übersetzungen because they're not any translations, but the translations of this bank.
In terms of dem Englischen und dem Französischen: The Duden-Grammatik describes several other usage scenarios of the definite article in §§ 387–400. One of them is material uniqueness (l.c., § 389), i.e., something is one of a kind (either absolutely, like the pope, or in the respective context, like the closest train station in the question Wo ist der Bahnhof?). Since there's only one English language, it seems comprehensible to use the definite article for languages.
However, in my opinion at least the article before Übersetzungen is not obligatory. I'm not entirely sure, however, about the articles regarding the languages. This brings us to the question of an alternative wording:
You ask why the sentence doesn't read Ich war für Übersetzungen aus Englisch-
en und Französisch- en zuständig. If the sentence could be written without these articles (despite the rule of uniqueness), you would at least need to omit the endings. The reason for this is maybe more complicated than one would expect:
Most substantival language denominations in German are derived from adjectives. However, there are two different kinds of substantivated language names (c.f. Duden-Grammatik, § 473):
- a lexical, endingless, rather substantival alternative (e.g., das Deutsch)
- and a syntactical, rather adjectival one with weak adjective endings (e.g., das Deutsche)
There are subtle usage tendencies for each form. In general the form with weak adjective endings is used, while in composita like das Schriftdeutsch and phrases like das Deutsch des Biedermeiers the endingless form is preferred.
So in your example you try to use the adjectival form without articles. However, without definite article, strong endings would be required for syntactical reasons (which is impossible—there is nothing like *Übersetzungen aus Deutschem und Englischem). Thus, we need to switch to the endingless substantival form here as well:
?Ich war für Übersetzungen aus Englisch und Französisch zuständig.
The reason for my doubts is mainly that if that sentence wasn't plainly wrong, it would probably have another reason: it's often possible to omit the article in enumerations (c.f. this answer). *Ich war für Übersetzungen aus Englisch zuständig is not a grammatical sentence in my opinion—and provided that it is indeed not, it is because of the rule of uniqueness (see above).
In any case, I would strongly prefer the version from the novel, especially in a literary context. Aus dem Englischen [übersetzt] is a fixed phrase, e.g., on the title page of a novel.