The accent is not unusual for a person whose native language is German. Actually the accents Germans have when speaking English are influenced by a lot of things - not the least by from which region of Germany they come and how they acquired their language skills.

Von Braun came from an upper class background and got higher Education including English and French from an early age.

The accent of von Braun is a not untypical example of an educated German with higher English skills (so he was described here https://www.google.de/amp/s/www.nzz.ch/amp/wochenende/schwerpunkt/mondlandung-raketenkonstrukteur-mit-nazi-vergangenheit-ld.1495321 and here https://www.google.de/amp/s/www.spiegel.de/politik/kolumbus-des-alls-a-10d8a63b-0002-0001-0000-000031971881-amp by German media). To my German ears the recording you posted sounds as if he was reading an English script with some words he simply pronounced as if they were written in German.

Note that it is reported von Braun became ever more fluent in English and joked about his accent:

‚He warmed up his 1950s audiences for speeches outside the South by apologizing “for my accent,” then grinning and adding, “I’m from Alabama.”‘ (Bob Ward https://appel.nasa.gov/2006/04/01/wernher-von-braun-lessons-taught-and-learned/).