These are the known Third Reich recordings of the song:
27.02.1940 | Rundfunk | Chor des RAD, Soldatenchor, Musikkorps des Regiments "General Göring", Ltg. Paul Haase
18.05.1940 | Rundfunk | Männerchor, Blasorchester
24.06.1940 | Grammophon | Soldatenchor, Reichsmusikzug des RAD, Ltg. Herms Niel
24.09.1940 | Lindström (Odeon) | Waldo-Favré-Chor, Musikkorps des Wachbataillons Berlin, Ltg. Guido Grosch
02.07.1941 | Grammophon | Soldatenchor, Musikkorps des Wachbataillons Berlin, Ltg. Guido Grosch
* the version with siren sound
All five performances used the brass band arrangement by Willi Lachner (with some differences in tempo; the 27.02.1940 and the 24.06.1940 versions are particularly slow).
It's likely that additional radio (Rundfunk) recordings were made, but there's no info on them so far (one can search for paratroopers' scenes in German wartime newsreels and check if there are different recordings of the song there; as I wrote before, newsreels used radio recordings).
A version with new lyrics was recorded for the 1942 film "
Sprung in den Feind".
The song can be also heard (as instrumental) in the 1944 film "Fallschirmjäger - Sturmsoldaten der Luft".
After WWII, the song was further recorded by the Bundeswehr (oddly enough, sometimes with Lincke's march "Märkische Heide" as its intro) and was falsely presented in several publications as a piece from 1936 or 1938, i.e. peacetime. Which was whitewashing rubbish. It's a wartime song, copyrighted and first recorded in 1940, which replaced the unsuccessful 1939 "Lied der Fallschirmjäger" by Kuntzen and Theilmann as the main paratroopers' song (note that the 1944 film "Fallschirmjäger - Sturmsoldaten der Luft" starts with a scene from the 1939 film "Fallschirmjäger", but this time set to the music of Schäfer's "Das Lied der Fallschirmjäger", while the 1939 film contained the Kuntzen-Theilmann's song there).
Cheers,
Ivan