I just installed TWOM (it calls itself The Soundtrack edition), through GOG Galaxy, but when I hit play, nothing happens. I say nothing, but the play button turns from purple to black for a few seconds, but that's it. I've tried uninstalling and re-downloading and installing again, to no avail.
Anyone have any idea what's going on?

Same issue here... did you ever manage to solve it?

Edit:

For anyone else facing this issue, it seems to be that the GOG installer screws up when trying to update the directX files required.

If you install them via Microsoft's installers, then it should work as intended - at time of writing I got a link to DirectX End-User Downloader through this thread which was helpful. Should apply to other games as well
https://answers.microsoft.com/en-us/windows/forum/games_windows_10/directx-9-on-windows-10/7b7cec82-4a6d-4bb8-b7a5-3889aae764c1

Hi, I seem to have similar problem. When I press Play button the game sort of starts but it is only a black screen and after couple of seconds I get a prompt message that the Program failed and do I want to wait or close it. I tried the tip with direct X but the installer says that the version is up to date.

Does anybody have other ideas?

I just installed it using Galaxy on win 10 64 bit and it works fine for me

Not sure it will be the same with steam version, but see if you have it.

In the TWoM there should be 2 additional exe file.
1. Located in the Win32 folder
2. Located in the x64 folder

Try them out, from experience the exe file in the Win32 folder usually work.