Can I remove the .nomedia file from Google Music downloaded music folder? I'm running version 4.1.511 of the Google Music app.
When it downloads songs for offline use they are stored in 


  /sdcard/Android/data/com.google.android.music/cache/music


This folder has a .nomedia file in it which prevents the media scanner from scanning the files. This means other music apps that I run don't see the songs?

Is there any problem with removing the .nomedia file from this directory so that the files can be scanned and picked up by other media players?
The .nomedia file is for the reason you stated. You can safely remove the file and other apps will be able to pick up the medias in that folder.

In any case, if there is any problem you can easily create that file again. It can be a empty file with the name .nomedia. You can use programs like ES File explorer to explore to that directory and create the file.

EDIT:
As per the comments below which contains very useful information:


Your Music apps may recreate the .nomedia file when you launch it.
Other players will only be able to play the music if only the musics are DRM free.