This might be of some interest: In the article Populäre und traditionelle Lieder. Historisch-kritisches Liederlexikon reference is made to a Swedish version. It states (translated): Of note is the obvious relationship with a Swedish children's song (noted before 1886): the shot bird is in this foreign-language parallel song a raven, which is at the end eaten by the hunter.

The Swedish version is at: http://www.liederlexikon.de/lieder/auf_einem_baum_ein_kuckuck/editiond

The lyrics are similar to the German, including the Simsalabim bit. Roughly translated, it contains the following verses:

1. Raven sits on tree

2. A hunter comes along

3. He shot the raven.

4. He took the raven home.

5. He put the raven in a pot.

6. He ate the raven.        
                 
Cheers, Don