https://bugs.winehq.org/show_bug.cgi?id=50778
--- Comment #2 from Alex Folland lexlexlex@gmail.com --- Your comment and Fabian's comment in the bug report you linked pointed me in the right direction for a personal-use workaround.
I have an updated copy of Windows 7 64-bit from which I can use system DLLs. I copied the following DLLs from my Windows drive to ~/.wine/drive_c/windows/system32/ and renamed them to lower-case versions, then set them to be used as overrides in winecfg:
-C:\Windows\System32\UIRibbon.dll -C:\Windows\System32\UIRibbonRes.dll -C:\Windows\System32\uxtheme.dll -C:\Windows\System32\propsys.dll
Once this was done, TortoiseMerge opened in wine without any apparent issues.
This workaround involves using proprietary non-free DLLs, so it's not a permanent solution for everyone, but it works for my personal usage.