How much improvisation are the instrumentalists allowed to do when recording music?

It would depend on what was being recorded and, most importantly, who was in charge of the recording. That would apply to any kind of music.

Anything from absolutely none to 100% improvised for that particular take.

It depends entirely on the music that is being recorded.

most songs have numerous writers

If you are at a session synchronizing the sound track to a video, you will be required to play everything exactly as written.

 I do a lot of improvisation.  If the copyright has expired, there is nobody to tell me what I can't do.  A movie score is probably copyrighted and the composer would not want it changed.