Runing it under Windows 10 64 here and the music plays fine.
I'm also running it on Windows 10 64 Home, on an Alienware 17 R4.
I'm pulling my hair out.
I've gone through everything and can't understand why I'm not getting any music.
Edit: I uninstalled, restarted my laptop and reinstalled Warlords 3.
I now have music but for some reason whenever I start a new turn the next track starts as if the first couple of seconds of the track have been clipped. Also after a track has finished it begins playing the next, stops and returns to the previous track.
Also whenever I open the game I receive a message saying "Warning. You are low on disk space. This may affect gameplay".
I am definitely not low on disk space.
For now and since the beginnng the music in this game has worked fine for me. But I suspect that the reason for your issues is the built in sound wrapper added by GOG to make the old music system working, If I am not wrong It was based in redbook audio or so, music track embebbed as a traditional music CD.
The wrapper makes the game read music files directly from the game folder as a modern workaround.
Some users have problems under windows 10 with the winimm.dll file under windows 10 osassionally.
I believe it is a file called winimm.dll and its configurator or ini file.
I have some problems with this system in other game like Imperialism (jumping tracks, stopping the music and not playing again etc), but as I say Warlords seems to work fine.
The possible solution could be try other/older version of the file but not sure.
About the message you are commenting about, just ignore it. Is a system message about incompatibility that dissapear if you run the game in windows 98 compatibility but I do not recommend do it because It can cause glitches and the game is already prepared to run in windows 7/10 as is.
The message is harmless. Just residual. It does not affect the game. But you can make a tr and run the game in compatibility to see if that is better for your music problem.