How far did you get into the game? It should be noted that some users told me that they were confused by the game's wordless and buttonless start screen. If that is not the issue here and there is a major issue with Wine compatibility, I may create and upload a Linux build (though I will not be able to test it).
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I couldn't start at all! ^^' (maybe because of the GUI or because I overlooked something)
So if I understand what you say maybe it's not Wine compatibility (not sure though!) and in that case a little bit more instructions would help!
If you are on Unity sometimes it happens the videos no longer work (if you have some) on export but everything should work just fine otherwise.
If there is no issue, starting the game works like so: launch the application, then hold space when the white screen with the rotating sun and red ring appears. Holding space for a second or two will begin the game.
However, if the described screen does not appear or if pressing space does nothing, then there is certainly a bug.