And the known workaround can work too (without rebuilding wine). It's linked in that bug. Example for Outlaws:
https://gist.github.com/shmerl/baa5367fce111e6999a4

I've tried installing the game in the latest version of CX, as well as in Wineskin with the latest CX 15 and CX 14 engines. No joy. I'll try out shmerl's shell script and see what happens.

Adjust it to your game first. Find all binaries which use WINMM.dll (or whatever case it might be). Patch only them. (Do binary grep for winmm.dll but case insensitive to find all such files).

The binaries that use winmm.dll? You mean, the EXE's that specifically call on it?

Both EXE and DLL files in the game. In case of Outlaws it were only some DLLs as you can see in the script.

Well, the files are binary, so opening them with a text editor while not impossible is impractical. The easiest way to find which files have this string is to open a terminal in the directory of the game (typically installed somewhere in ~/.wine/drive_c/GOG GAMES etc.) and run grep, as

grep -i winmm.dll *{DLL,EXE}

(the option -i does case-insensitive search, followed by the search string and the list of files)

See what igrok said above. grep can search in binary files. It's a pretty powerful and useful tool.

Game: Vampire: The Masquerade - Redemption
Installer MD5: 96c19b0ce487e53dd8fe24ef963d31fc setup_vampire_the_masquerade_redemption_2.0.0.3.exe
WineHQ AppDB link: https://appdb.winehq.org/objectManager.php?sClass=application&iId=1772
CodeWeavers link: https://www.codeweavers.com/compatibility/crossover/vampire-the-masquerade-redemption

Distro: Linux Mint 17.3 Cinnamon 64bit
Kernel version: 3.19.0-58
Graphics card: Nvidia GeForce GTX 750Ti
Graphics driver & version: Proprietary 367.27
Wine version(s) tested: Wine Staging 1.9.13 via PlayOnLinux, CrossOver 15.1.0

Install notes: Just install & play.
If using PlayOnLinux, run the game from/create a shortcut for "Vampire.exe"
How well does it run: Almost perfect
Details: Have only played for a short while but encountered no major problems - only that it messes around with display brightness & doesn't reset it on exiting (just open NVIDIA X Server Settings after playing to return it to normal). Tested with virtual desktop enabled. Game will hang on exit if CSMT is enabled.

Edit: I finally cracked it. I'll put up the requisite post in a few minutes.

Pure Wine: just tested five GOG games in here (Caesar III, Haegemonia, Railroad Tycoon 2) and a couple of oldies of mine (Remington Top Shot/Gobliiins IV) and they all work amazingly perfect ;)

Game: Parkan: The Imperial Chronicles

Installer MD5s: 9e842b312774640b1184056f3c05003f setup_parkan_the_imperial_chronicles_2.0.0.6.exe

WineHQ AppDB link: https://appdb.winehq.org/objectManager.php?sClass=application&iId=9631
CodeWeavers link: n/a

Distro: Mac OS X El Capitain 10.11.5
Kernel version: Darwin 15.4.0 (Build 15F34)
Graphics card: Nvidia GeForce 320M / Intel HD 5000 Graphics
Graphics driver & version: OS Default
Wine version(s) tested: WINE 1.9.11 / wine-staging 1.9.12 via Wineskin

Install notes:
1. To avoid MCICD errors / crashes on startup:
- locate NAV32.DLL, NGI32.DLL, WALK3D.DLL, and PARKAN.EXE in the game's root folder
- for each of the above four files, do the following:
- open it in TextEdit
- search for "winmm"; in each of the four files, there should only be one instance of that text string
- replace "WINMM.DLL" with "WIN32.DLL"
- locate winmm.dll in the game's root folder
- rename winmm.dll as win32.dll

(Special thanks to herecomesthe2000 for this tip, one of several apparent workarounds for bug #37983; they originally posted this as a step to enable music in X-Wing '95 and TIE-Fighter '95 under WINE in OS X. Apparently GOG already applied this to X-Wing vs. TIE-Fighter. Special thanks also go to shmerl and igrok for their help in finding the files, and leading me to this solution.)

2. To get movies working:
- install the following, in the order below:
1. msxml3 (winetricks)
2. wmp9 (winetricks)
3. Ligos Indeo Video Codecs (available here: http://www.free-codecs.com/download/indeo_codec.htm)

When installing the Ligos Indeo codecs, ask for a Custom install, and disable the web browser plugin option.

(Installing wmp9 first, solves an error in the Ligos Indeo Video installer with regsvr32, where Ivfsrc.ax could not be properly registered; this is an issue I've seen with installations of Indeo Video on other games I've tried in WINE.)

- set ddr=gdi in winetricks

How well does it run: Almost perfectly...Both FPS and space combat seem to work as expected. There's a slight cosmetic glitch with the Exit screen that appears when pressing ESC during the FPS levels, but it's very minor.

Details: The game is hard-locked at 640 x 480; there is an option to play fullscreen without the HUD in the FPS levels, but it's not recommended as you lose the health and environmental information seen with the HUD enabled. There is hardware acceleration support in the game's FPS levels and it did work during my testing. There is also apparently no way to save the game during the FPS levels. (!) On OS X, enabling the Native Mac Driver is recommended as otherwise the mouse cursor may disappear during gameplay.

There is also no way to specify settings or look up controls from within the game; you need to go to the external SETUP.EXE, or consult the manual for controls.

Game: Sword of the Stars: The Pit - Gold Edition (Updated GOG Gold Edition)

Installer MD5s: ce1c4d9363544121f14a37578c2b667f setup_sots_the_pit_gold_2.7.0.13.exe

WineHQ AppDB link: https://appdb.winehq.org/objectManager.php?sClass=version&iId=32195
CodeWeavers link: https://www.codeweavers.com/compatibility/crossover/sword-of-the-stars-the-pit

Distro: Mac OS X El Capitain 10.11.5
Kernel version: Darwin 15.4.0 (Build 15F34)
Graphics card: Intel HD 5000 Graphics
Graphics driver & version: OS Default
Wine version(s) tested: wine-staging 1.9.12 via Wineskin

Install notes:
The game runs and installs perfectly out of the box; even the GOG installer's .NET 4 and XNA4 installers work without any incident. No extra winetricks or earllier installations of .NET 2/3/etc. are necessary. IIRC, the patches that allow for this have been in the staging branch since 1.8. AFAIK those patches haven't been yet merged back into the mainline 1.8.x or 1.9.x builds.

How well does it run: Almost perfectly...saving/loading games as well as all of the DLC (including the new Medic) work as advertised.

Details: Tested with the Native Mac Driver enabled; the game won't display properly without it. The game also always starts up with a .NET Configuration Parser error message, but that doesn't seem to have any impact on the game itself. Unfortunately, my Logitech F310 didn't work in the game.

Can you properly switch to full screen mode? I set it up in 1.9.13-staging via PoL a few days ago, it works perfectly other the fact it freezes up when changing to full screen. Not a big deal since it looks great in a maximized window, but i'm just curious.