I was able to install JP3 on wine but it won't play any videos, and when I click on new game, I get a blank text box with just the option "Cancel". Then another text box with no text or buttons. I can push escape, and then I get the message "Sorry, the game cannot continue without the disc" with the options "continue" and "main menu".

I used the offline installer, but I get the same problem when using the gog installer (and installing galaxy first)

I got it working on Wine. Installed via Lutris, so I'm using WIne version lutris-5.7.11, but I don't think it matters.
In Wine configuration, under Drives, I added drive D:, under Show Advanced options set it to type CD-ROM, and set the path to [install directory]/drive_c/GOG Games/Journeyman Project 3 - Legacy of Time/data/.

I've also edited the registry in HKEY_LOCAL_MACHINE\Software\Wow6432Node\GOG.com\GOGJOURNEYMAN3 and set GOGGAMEDLL to goggame.dll because that value was empty, but I'm not sure if it's necessary for the game to work.

The view scrolling in game is still very fast, but I also get that in Windows and don't know yet how to fix it.