RAW, you can speak any language you know, but it is unclear as to whether you are disallowed from speaking Common, Elvish, and Sylvan.
The relevant section in wild shape (PHB 67) states:
... and your ability to speak [...] is limited to the capabilities of your beast form.
The relevant section in the sage advice compendium 2019 (page 3):
Can a Circle of the Moon druid speak the languages it
knows while in the form of an elemental?
Yes, since the
elementals listed in Elemental Wild Shape can speak.
A literal interpretation (RAW) of Wild Shape could
reasonably lead you to think that transformed druids
can speak only languages that appear in an elemental’s
stat block, but the intent (RAI) is that druids retain their
knowledge, including of languages, when they transform
and can speak the languages they know if an adopted form
So you retain your knowledge of all languages (this includes not knowing Giant Elk).
However, it is unclear whether it is a limitation of the capabilities of the beast form that prevents it from speaking the three languages or something else.
My opinion on the matter, as a DM
If it is the physical capabilities of the creature that prevent it from speaking those 3 languages (we see this in the real world with dolphins, for example, who speak "dolphin", but can also understand the humans, but cannot speak English), then it perhaps could be assumed that Giant Elks work the same way, and therefore are limited to not speaking any language.
If it is a mental thing that prevents the creatures from speaking the 3 languages, then it does not make sense that a new mind inhabiting a Giant Elk's body would be subject to the same limitation.
I, therefore, would either say that you can speak no languages, undercutting the RAW ruling, or could speak all of the ones you know, ignoring the potential 3 language restriction ruling.