There are two issues. The first is that the game complains that it can not load global data. To get around this, one has to:

1) copy contents of DAO directory (located under wherever Liberation Day was installed) to $WINEPREFIX/drive_d/windows/system32
2) set dll override for each copied *.dll
3) launch command line under Wine and register
regsvr32 /s c:\windows\system32\DAO350.DLL
regsvr32 /s c:\windows\system32\MSJET35.DLL
4) install vb6run and jet40 with winetricks

The second issue (and that seems to be shared with original Fallen Haven) is that though the game now launches, sprites are not rendered correctly. Results are partially random and may range from black screen to only ground plane visible to most of the stuff (units, structures, trees etc.) visible, but not all. This may be related to 256 colour mode the game uses, but my experiments with Xephyr to force 8bit colour depth were unsuccessful (black screen).

So I am leaving this bit of knowledge here for sake of science and for future generations. Perhaps one day somebody will crack the second issue too.

I make this post to people who still want to play this game in 2024.

Experimental ntsync feature fixes most graphical glitches and game became playable.
There are still minor glitches like some text shifted to left. And there still ~10% chance for that graphic glitch to reappear.

Steps to reproduce:

1) Get ntsync module running. It's not enabled in default kernel, google how to do it on your distro!
On arch install ntsync-dkms, ntsync-common, ntsync-header from aur.
#modprobe ntsync

Now you should have /dev/ntsync device.

2) Get wine with ntsync support. Sometimes its called fastsync (not fsync).
I used wine-tkg-git from Frogging-Family on github
Read wine-tkg-git/customization.cfg twice.

Change installation location
_nomakepkg_prefix_path="/opt"
_EXTERNAL_INSTALL="true"

Enable ntsync
_use_ntsync="true"

and disable conflicting options
_use_esync="false"
_use_fsync="false"

Build it and install.
On arch just
>make -risc

3) Make your life easier(Optional)
I use bottles to manage my wine prefixes and wine versions.
You can add wine built in step 2 to list of runners in bottles.
Go to /opt and copy wine-tkg-staging-ntsync-something to ~/.local/share/bottles/runners/
Making just a symlink dont work because bottles want rw rights on runner files.

4) Do all steps from first post!

5) Other minor fixes
Run winecfg, enable virtual desktop and set it to 1024x768. This fixes cursor positioning.
Install cnc-ddraw either via winetricks or dependencies tab in bottles.
Delete outdated EXE\ddraw.dll. Run game once. Close it. This should create new file EXE\ddraw.ini
In EXE\ddraw.ini set
fixpitch=true
This should fix tilted text

Thats all. There are still minor bugs:
Graphic glitch has low chance to reappear - just reload save.
Some text shifted to left - I had no luck fixing this.