Dictionaries of English list the word wombat as being of “native origin”
Also note the German word Maus, a reduced form of the earlier (Grund)maus.
Obviously, the German morpheme maus can be identified with the English bat. The original meaning of both was ‘any smallish rodentlike critter’. In English, for whatever reason, the flying bat eventually came to be seen as the prototypical form of bat; in German, on the other hand, the ground mouse came to be seen as the prototype. English obviously borrowed the word maus from German sometime after it had come to mean simply ‘mouse’, and also after the Great Vowel Shift had occurred.
Now, comparing German Beutelmaus with English wombat, if we discard the morphemes maus and bat, we are left with Beutel = ‘pouch’ and wom = ? . Well, this is not such a hard puzzle to solve. All we need to do is find a morpheme resembling wom in shape and Beutel in meaning. The morpheme we seek is clearly womb, which in earlier English meant any body cavity, and which would naturally combine with bat to form wombat, not *wombbat.
Beutelmaus is an evident loan translation of English wombat into German; we all know how fond the German language is of loan translations. Of course, it sometimes borrows words, directly, as well, and in fact there exists a noun der Wombat in German. The situation is exactly analogous to that of Fernsprecher/Telefon.
To sum up; wombat is not of Australian aboriginal origin. It is a purely English descriptive compound whose originally transparent meaning was ‘pouch rodent’.